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1,200 |
Another homogeneous, non-bihomogeneous Peano continuum
|
math.GN
|
K. Kuperberg found a locally connected, finite-dimensional continuum which is
homogeneous but not bihomogeneous. We give a similar but simpler example. Like
previous constructions, the example is locally a Cartesian product of Menger
spaces. The new idea is to choose a fundamental group in which not every
element is conjugate to its inverse.
|
math
|
1,201 |
A hereditarily indecomposable tree-like continuum without the fixed point property
|
math.GN
|
A hereditarily indecomposable tree-like continuum without the fixed point
property is constructed. The example answers a question of Knaster and Bellamy.
|
math
|
1,202 |
More on sg-compact spaces
|
math.GN
|
The aim of this paper is to continue the study of sg-compact spaces, a
topological notion much stronger than hereditary compactness. We investigate
the relations between sg-compact and $C_2$-spaces and the interrelations to
hereditarily sg-closed sets.
|
math
|
1,203 |
What is supertopology?
|
math.GN
|
We discuss the problem of finding an analogue of the concept of a topological
space in supergeometry, motivated by a search for a procedure to compactify a
supermanifold along odd coordinates. In particular, we examine the topologies
naturally arising on the sets of points of locally ringed superspaces, and show
that in the presence of a nontrivial odd sector such topologies are never
compact. The main outcome of our discussion is that not only the usual
framework of supergeometry (the theory of locally ringed spaces), but the more
general approach of the functor of points, need to be further enlarged.
|
math
|
1,204 |
Some covering properties of the $α$-topology
|
math.GN
|
Recently, Mr\v{s}evi\'{c} and Reilly discussed some covering properties of a
topological space and its associated $\alpha$-topology in both topological and
bitopological ways. The main aim of this paper is to investigate some common
and controversial covering properties of $\cal T$ and ${\cal T}^{\alpha}$.
|
math
|
1,205 |
Unification approach to the separation axioms between $T_0$ and completely Hausdorff
|
math.GN
|
The aim of this paper is to introduce a new weak separation axiom that
generalizes the separation properties between $T_1$ and completely Hausdorff.
We call a topological space $(X,\tau)$ a $T_{\kappa,\xi}$-space if every
compact subset of $X$ with cardinality $\leq \kappa$ is $\xi$-closed, where
$\xi$ is a general closure operator. We concentrate our attention mostly on two
new concepts: kd-spaces and $T_{1/3}$-spaces.
|
math
|
1,206 |
Idealization of some weak separation axioms
|
math.GN
|
An ideal is a nonempty collection of subsets closed under heredity and finite
additivity. The aim of this paper is to unify some weak separation properties
via topological ideals. We concentrate our attention on the separation axioms
between $T_0$ and $T_2$. We prove that if $(X,\tau,{\cal I})$ is a
semi-Alexandroff $T_{\cal I}$-space and $\cal I$ is a $\tau$-boundary, then
$\cal I$ is completely codense.
|
math
|
1,207 |
A remark on $β$-locally closed sets
|
math.GN
|
The aim of this note is to show that every subset of a given topological
space is the intersection of a preopen and a preclosed set, therefore
$\beta$-locally closed, and that every topological space is $\beta$-submaximal.
|
math
|
1,208 |
An answer to a question of Coleman on scattered sets
|
math.GN
|
The aim of this paper is to show that every scattered subset of a
dense-in-itself semi-$T_D$-space is nowhere dense. We are thus able to answer a
recent question of Coleman in the affirmative. In terms of Digital Topology, we
prove that in semi-$T_D$-spaces with no open screen, trace spaces have no
consolidations.
|
math
|
1,209 |
On p-closed spaces
|
math.GN
|
In this paper we will continue the study of p-closed spaces. This class of
spaces is strictly placed between the class of strongly compact spaces and the
class of quasi-H-closed spaces. We will provide new characterizations of
p-closed spaces and investigate their relationships with some other classes of
topological spaces.
|
math
|
1,210 |
Contra-semicontinuous Functions
|
math.GN
|
The aim of this paper is to introduce and study the concept of a
contra-semicontinuous function and further investigate the class of strongly
$S$-closed spaces. We obtain some new decompositions of generalized continuous
functions.
|
math
|
1,211 |
G.Λ_s-sets and G.V_s-sets
|
math.GN
|
In this paper we define the concepts of $g.\Lambda_s$-sets and $g.V_s$-sets
and we use them in order to obtain new characterizations of semi-T_1-,
semi-R_0- and semi-T_{1/2}-spaces.
|
math
|
1,212 |
Locally 1-to-1 maps and 2-to-1 retractions
|
math.GN
|
This paper considers the question of which continua are 2-to-1 retracts of
continua.
|
math
|
1,213 |
Exactly k-to-1 maps: from pathological functions with finitely many discontinuities to well-behaved covering maps
|
math.GN
|
Many mathematicians encounter k-to-1 maps only in the study of covering maps.
But, of course, k-to-1 maps do not have to be open. This paper touches on
covering maps, and simple maps, but concentrates on ordinary k-to-1 functions
(both continuous and finitely discontinuous) from one metric continuum to
another. New results, old results and ideas for further research are given; and
a baker's dozen of questions are raised.
|
math
|
1,214 |
On a stronger form of hereditary compactness in product spaces
|
math.GN
|
The aim of this paper is to continue the study of sg-compact spaces. The
class of sg-compact spaces is a proper subclass of the class of hereditarily
compact spaces. In our paper we shall consider sg-compactness in product
spaces. Our main result says that if a product space is sg-compact, then either
all factor spaces are finite, or exactly one factor space is infinite and
sg-compact and the remaining ones are finite and locally indiscrete.
|
math
|
1,215 |
A note on Saleh's paper `Almost continuity implies closure continuity'
|
math.GN
|
Recently, Saleh claimed to have solved `a long standing open question' in
Topology; namely, he proved that every almost continuous function is closure
continuous (= $\theta$-continuous). Unfortunately, this problem was settled
long time ago and even a better result is known.
|
math
|
1,216 |
An answer to a question of Pyrih
|
math.GN
|
We answer a recent question of Pyrih by proving that a topological space
$(X,\tau)$ is open-normal if and only if it is extremally disconnected.
|
math
|
1,217 |
Survey on preopen sets
|
math.GN
|
The aim of this survey article is to cover most of the recent research on
preopen sets. I try to present majority of the results on preopen sets that I
am aware of.
|
math
|
1,218 |
Between ${\cal A}$- and ${\cal B}$-sets
|
math.GN
|
The aim of this paper is to introduce the class of ${\cal A}{\cal B}$-sets as
the sets that are the intersection of an open and a semi-regular set. Several
classes of well-known topological spaces are characterized via the new concept.
A new decomposition of continuity is provided.
|
math
|
1,219 |
Topological properties defined in terms of generalized open sets
|
math.GN
|
This paper covers some recent progress in the study of sg-open sets,
sg-compact spaces, N-scattered spaces and some related concepts. A subset $A$
of a topological space $(X,\tau)$ is called sg-closed if the semi-closure of
$A$ is included in every semi-open superset of $A$. Complements of sg-closed
sets are called sg-open. A topological space $(X,\tau)$ is called sg-compact if
every cover of $X$ by sg-open sets has a finite subcover. N-scattered space is
a topological spaces in which every nowhere dense subset is scattered.
|
math
|
1,220 |
Extremally $T_1$-spaces and Related Spaces
|
math.GN
|
The aim of this paper is introduce and initiate the study of extremally
$T_1$-spaces, i.e., the spaces where all hereditarily compact $C_2$-subspaces
are closed. A $C_2$-space is a space whose nowhere dense sets are finite.
|
math
|
1,221 |
On locally LC-spaces
|
math.GN
|
A topological space $(X,\tau)$ is called a locally LC-space if every point of
$X$ has a neighborhood $U$ such that every Lindel\"{o}f subset of $(U,\tau|U)$
is a closed subset of $(U,\tau|U)$. The aim of this paper is to continue the
study of locally LC-spaces.
|
math
|
1,222 |
Idealization of Ganster-Reilly decomposition theorems
|
math.GN
|
In 1990, Ganster and Reilly proved that a function is continuous if and only
if it is precontinuous and LC-continuous. In this paper we extend their
decomposition of continuity in terms of ideals. We show that a function $f
\colon (X,\tau,{\cal I}) \to (Y,\sigma)$ is continuous if and only if it is
pre-I-continuous and I-LC-continuous. We also provide a decomposition of
I-continuity.
|
math
|
1,223 |
p-topological and p-regular: dual notions in convergence theory
|
math.GN
|
The natural duality between "topological" and "regular," both considered as
convergence space properties, extends naturally to p-regular convergence
spaces, resulting in the new concept of a p-topological convergence space.
Taking advantage of this duality, the behavior of p-topological and p-regular
convergence spaces is explored, with particular emphasis on the former, since
they have not been previously studied. Their study leads to the new notion of a
neighborhood operator for filters, which in turn leads to an especially simple
characterization of a topology in terms of convergence criteria. Applications
include the topological and regularity series of a convergence space.
|
math
|
1,224 |
Homotopy classes of maps between Knaster continua
|
math.GN
|
By a Knaster continuum we understand the inverse limit of copies of [0,1]
with open bonding maps. We prove that for any two Knaster continua K_1 and K_2,
there are 2^\aleph_0 distinct homotopy types of maps of K_1 onto K_2 that map
the endpoint of K_1 to the endpoint of K_2.
|
math
|
1,225 |
Symmetric Products and Q-manifolds
|
math.GN
|
An example is given of a compact absolute retract that is not a Hilbert cube
manifold but whose second symmetric porduct is the Hilbert cube. A factor
theorem is given for nth symmetric product of the cartesian product of any
absolute neighborhood retract with the Hilbert cube. A short proof is included
of the known fact that symmetric products preserve the property of being a
compact Hilbert cube manifold (the theorem is proved here for all Hilbert cube
manifolds).
|
math
|
1,226 |
Asymptotic topology
|
math.GN
|
We establish some basic theorems in dimension theory and absolute extensor
theory in the coarse category of metric spaces. Some of the statements in this
category can be translated in general topology language by applying the Higson
corona functor. The relation of problems and results of this `Asymptotic
Topology' to Novikov and similar conjectures is discussed.
|
math
|
1,227 |
On spaces Baire isomorphic to the powers of the real line
|
math.GN
|
We characterize AE(0)-spaces that are Baire isomorphic to the powers of the
real line.
|
math
|
1,228 |
Compactifications and universal spaces in extension theory
|
math.GN
|
We show that for each countable simplicial complex P the following conditions
are equivalent: (1) $P \in AE(X)$ iff $P \in AE(\beta X)$ for any space X; (2)
There exists a P-invertible map of a metrizable compactum X with $P \in AE(X)$
onto the Hilbert cube.
|
math
|
1,229 |
On some dimensional properties of 4-manifolds
|
math.GN
|
It is shown, under the assumption of Jensen's principle $\lozenge$, that if
for a complex L with $[L] \geq [S^{4}]$ there exists a metrizable compactum
whose extension dimension is L, then there exists a differentiable, countably
compact, perfectly normal and hereditarily separable 4-manifold whose extension
dimension is also [L].
|
math
|
1,230 |
Topological characterization of torus groups
|
math.GN
|
Topological characterization of torus groups is given.
|
math
|
1,231 |
Compact groups and absolute extensors
|
math.GN
|
We discuss compact Hausdorff groups from the point of view of the general
theory of absolute extensors. In particular, we characterize the class of
simple, connected and simply connected compact Lie groups as AE(2)-groups the
third homotopy group of which is $\text{\f Z}$. This is the converse of the
corresponding result of R. Bott.
|
math
|
1,232 |
Universal metric spaces and extension dimension
|
math.GN
|
For any countable $CW$-complex $K$ and a cardinal number $\tau\geq\omega$ we
construct a completely metrizable space $X(K,\tau)$ of weight $\tau$ with the
following properties: $\e X(K,\tau)\leq K$, $X(K,\tau)$ is an absolute extensor
for all normal spaces $Y$ with $\e Y\leq K$, and for any completely metrizable
space $Z$ of weight $\leq\tau$ and $\e Z\leq K$ the set of closed embeddings
$Z\to X(K,\tau)$ is dense in the space $C(Z,X(K,\tau))$ of all continuous maps
from $Z$ into $X(K,\tau)$ endowed with the limitation topology. This result is
applied to prove the existence of universal spaces for all metrizable spaces of
given weight and with a given cohomological dimension.
|
math
|
1,233 |
Extension dimension and refinable maps
|
math.GN
|
Extension dimension is characterized in terms of $\omega$-maps. We apply this
result to prove that extension dimension is preserved by refinable maps between
metrizable spaces. It is also shown that refinable maps preserve some
infinite-dimensional properties.
|
math
|
1,234 |
Topological groups: where to from here?
|
math.GN
|
This is an account of one man's view of the current perspective of theory of
topological groups. We survey some recent developments which are, from our
viewpoint, indicative of the future directions, concentrating on actions of
topological groups on compacta, embeddings of topological groups, free
topological groups, and `massive' groups (such as groups of homeomorphisms of
compacta and groups of isometries of various metric spaces).
|
math
|
1,235 |
Spaces having a small diagonal
|
math.GN
|
We obtain several results and examples concerning the general question ``When
must a space with a small diagonal have a G_delta-diagonal?". In particular, we
show (1) every compact metrizably fibered space with a small diagonal is
metrizable; (2) there are consistent examples of regular Lindelof (even
hereditarily Lindelof) spaces with a small diagonal but no G_delta-diagonal;
(3) every first-countable hereditarily Lindelof space with a small diagonal has
a G_delta-diagonal; (4) assuming CH, every Lindelof Sigma-space with a small
diagonal has a countable network; (5) whether countably compact spaces with a
small diagonal are metrizable depends on your set theory; (6) there is a
locally compact space with a small diagonal but no G_delta diagonal.
|
math
|
1,236 |
Characterizing the topology of pseudo-boundaries of Euclidean spaces
|
math.GN
|
We give a topological characterization of the n-dimensional pseudo-boundary
of the (2n+1)-dimensional Euclidean space.
|
math
|
1,237 |
Topological AE(0)-groups
|
math.GN
|
We investigate topological AE(0) -groups class of which contains the class of
Polish groups as well as the class of all locally compact groups. We establish
the existence of an universal AE(0) -group of a given weight as well as the
existence of an universal action of AE(0) -group of a given weight on a AE(0)
-space of the same weight. A complete characterization of closed subgroups of
powers of the symmetric group is obtained. It is also shown that every AE
(0)-group is Baire isomorphic to the product of Polish groups. These results
are obtained by using the spectral descriptions of AE(0)-groups which are
presented in Section 3.
|
math
|
1,238 |
Topological Representations of Posets
|
math.GN
|
Earlier an arbitrary poset $P$ was proved to be isomorphic to the collection
of subsets of a space $M$ with two closures which are closed in the first
closure and open in the other. As a space $M$ for this representation an
algebraic dual space $P^*$ was used. Here we extend the theory of algabraic
duality for posets generalizing the notion of an ideal. This approach yields a
sufficient condition for the collection of clopen subsets of a subset of $P^*$
(with respect to induced closures) to be isomorphic to $P$. Applying this
result to certain classes of posets we prove some representation theorems and
get a topological characterization of orthocomplementations.
|
math
|
1,239 |
Hereditary indecomposability and the Intermediate Value Theorem
|
math.GN
|
We show that hereditarily indecomposable spaces can be characterized by a
special instance of the Intermediate Value Theorem in their rings of continuous
functions.
|
math
|
1,240 |
A small transitive family of continuous functions on the Cantor set
|
math.GN
|
In this paper we show that, when we iteratively add Sacks reals to a model of
ZFC we have for every two reals in the extension a continuous function defined
in the ground model that maps one of the reals onto the other.
|
math
|
1,241 |
Universal metric spaces according to W. Holsztynski
|
math.GN
|
We show, following W. Holsztynski, that there exists a continuous metric d on
the set of real numbers R such that any finite metric space is isometrically
embeddable into (R,d).
|
math
|
1,242 |
A note on a question of R. Pol concerning light maps
|
math.GN
|
Let f:X -> Y be an onto map between compact spaces such that all
point-inverses of f are zero-dimensional. Let A be the set of all functions u:X
-> I=[0,1] such that $u[f^\leftarrow(y)]$ is zero-dimensional for all y in Y.
Do almost all maps u:X -> I, in the sense of Baire category, belong to A? H.
Toru\'nczyk proved that the answer is yes if Y is countable-dimensional. We
extend this result to the case when Y has property C.
|
math
|
1,243 |
The Roelcke compactification of groups of homeomorphisms
|
math.GN
|
Let X be a zero-dimensional compact space such that all non-empty clopen
subsets of X are homeomorphic to each other, and let H(X) be the group of all
self-homeomorphisms of X with the compact-open topology. We prove that the
Roelcke compactification of H(X) can be identified with the semigroup of all
closed relations on X whose domain and range are equal to X. We use this to
prove that the group H(X) is topologically simple and minimal, in the sense
that it does not admit a strictly coarser Hausdorff group topology.
|
math
|
1,244 |
Answering a question on relative countable paracompactness
|
math.GN
|
We answer a question of Yasui. Morever, we show that if a Tychonoff space Y
is countably 1-paracompact in every Tychonoff space X that contains Y as a
closed subspace then Y is linearly Lindelof.
|
math
|
1,245 |
Cardinal p and a theorem of Pelczynski
|
math.GN
|
We show that it is consistent that for some uncountable cardinal k, all
compactifications of the countable discrete space with remainders homeomorphic
to $D^k$ are homeomorphic to each other. On the other hand, there are $2^c$
pairwise non-homeomorphic compactifications of the countable discrete space
with remainders homeomorphic to $D^c$ (where c is the cardinality of the
continuum).
|
math
|
1,246 |
How weak is weak extent?
|
math.GN
|
A space X is star-Lindelof provided for every open cover U there is a finite
subset A of X such that St(A,U)=X. We show that a Tychonoff star-Lindelof space
can have arbitraryly big extent while the extent of a normal star-Lindelof
space can not be greater than c.
|
math
|
1,247 |
On spaces in countable web
|
math.GN
|
We show that a Tychonoff discretely star-Lindelof space can have arbitrarily
big extent and note that there are consistent examples of normal discretely
star-Lindelof spaces with uncountable extent.
|
math
|
1,248 |
Ljusternik-Schnirelmann Categories, Links and Relations
|
math.GN
|
This paper is concerned with some well-known Ljusternik-Schnirelmann
categories. We desire to find some links and relations among them. This has
been done by using the concepts of precategoty, T-collection and closure of a
category.
|
math
|
1,249 |
Duality of $κ$-normed topological vector spaces and their applications
|
math.GN
|
A duality of $\kappa$-normed topological vector spaces is defined and
investigated. For such spaces the analog of the Mackey-Arens theorem is proved.
There are investigated cases, when $\kappa$-normability of a topological vector
space implies its local convexity. There are given applications of $\kappa
$-normed spaces for resolutions of differential equations and for
approximations of functions in mathematical economy.
|
math
|
1,250 |
Extension dimensional approximation theorem
|
math.GN
|
Let $L$ be a countable CW-complex and $F\colon X\to Y$ be upper
semicontinuous $UV^{[L]}$-valued mapping of a paracompact space $X$ to a
complete metric space $Y$. We prove that if $X$ is a C-space of extension
dimension $\ed X \le [L]$, then $F$ admits single-valued graph approximations.
For $L=S^n$ our result implies well-known approximation theorem for
$UV^{n-1}$-valued mappings of $n$-dimensional spaces. And for $L=\{\rm point\}$
our theorem implies a theorem of Ancel on approximations of $UV^\infty$-valued
mappings of C-spaces.
|
math
|
1,251 |
Some mapping theorems for extensional dimension
|
math.GN
|
We present some results related to theorems of Pasynkov and Torunczyk on the
geometry of maps of finite dimensional compacta.
|
math
|
1,252 |
Movable categories
|
math.GN
|
The notion of movability for metrizable compacts was introduced by K.Borsuk
$[1]$. In this paper we define the notion of movable category and prove that
the movability of a topological space $X$ coincides with the movability of a
suitable category, which is generated by the topological space $X$ (i.e., the
category $W^X$, defined by S.Mardesic).
|
math
|
1,253 |
Some questions of equivariant movability
|
math.GN
|
In this paper the some questions of equivariant movability connected with
substitution of acting group $G$ on closed subgroup $H$ and with transitions to
spaces of $H$-orbits and $H$-fixed points spaces are investigated. In the
special case the characterization of equivariant-movable $G$-spaces is given.
|
math
|
1,254 |
Topological groups with several disconnectedness
|
math.GN
|
We investigate some properties of topological groups related to
disconnectedness or Archimedeanness. We prove or disprove the preservation of
those under operations as subgroups, quotients, products, etc.
Characterizations of non-Archimedeanness are obtained by using embedding into
universal groups. We also clarify the differences between the properties by
constructing miscellaneous Polish groups.
|
math
|
1,255 |
Topological semigroups and universal spaces related to extension dimension
|
math.GN
|
It is proved that there is no structure of left (right) cancelative semigroup
on $[L]$-dimensional universal space for the class of separable compact spaces
of extensional dimension $\le [L]$. Besides, we note that the homeomorphism
group of $[L]$-dimensional space whose nonempty open sets are universal for the
class of separable compact spaces of extensional dimension $\le [L]$ is totally
disconnected.
|
math
|
1,256 |
On the Maćkowiak-Tymchatyn theorem
|
math.GN
|
In this paper we give new proofs of the theorem of Ma\'{c}kowiak and
Tymchatyn that every metric continuum is a weakly-confluent image of some
one-dimensional hereditarily indecomposable continuum of countable weight. The
first is a model-theoretic argument; the second is a topological proof inspired
by the first.
|
math
|
1,257 |
Dense families of selections and finite-dimensional spaces
|
math.GN
|
A characterization of $n$-dimensional spaces via continuous selections
avoiding $Z_n$-sets is given, and a selection theorem for strongly
countable-dimensional spaces is established. We apply these results to prove a
generalized Ostrand's theorem and to obtain a new alternative proof of the
Hurewicz formula.
|
math
|
1,258 |
On the Semicontinuity in Product Spaces
|
math.GN
|
Let $X,Y$ be topological vector spaces or metric spaces, and let {$f:X\times
Y \to \Re $} be a real function lower semicontinuous in the first variable and
upper semicontinuous in the second one. It is proved that $f$ is globally
measurable. Sierpinski (1925) has been raised this question in the case
$X=Y=\Re $. This particular case was solved by Kempisty (1929). The actual
result has applications in Calculus of Variations.
|
math
|
1,259 |
On Finite-Dimensional Maps II
|
math.GN
|
Let $f\colon X\to Y$ be a perfect $n$-dimensional surjection of paracompact
spaces with $Y$ being a $C$-space. We prove that, for any $m\geq n+1$, almost
all (in the sense of Baire category) maps $g$ from $X$ into the $m$-dimensional
cube have the following property: $g(f^{-1}(y))$ is at most $n$-dimensional for
every $y\in Y$.
|
math
|
1,260 |
Continuous Selections and finite C-spaces
|
math.GN
|
Characterizations of paracompact finite $C$-spaces via continuous selections
are given. We apply these results to obtain some properties of finite
$C$-spaces. Factorization theorems and a completion theorem for finite $C$-
spaces are also established.
|
math
|
1,261 |
Euler characteristic of the configuration space of a complex
|
math.GN
|
A closed form formula (generating function) for the Euler characteristic of
the configuration space of $\scriptstyle n$ particles in a simplicial complex
is given.
|
math
|
1,262 |
An Algebraic and Logical approach to continuous images
|
math.GN
|
Continuous mappings between compact Hausdorff spaces can be studied using
homomorphisms between algebraic structures (lattices, Boolean algebras)
associated with the spaces. This gives us more tools with which to tackle
problems about these continuous mappings -- also tools from Model Theory. We
illustrate by showing that the \v{C}ech-Stone remainder $[0,\infty)$ has a
universality property akin to that of $N^*$; a theorem of Ma\'ckowiak and
Tymchatyn implies it own generalization to non-metric continua; and certain
concrete compact spaces need not be continuous images of $N^*$.
|
math
|
1,263 |
Sublinear and continuous order-preserving functions for noncomplete preorders
|
math.GN
|
We characterize the existence of a nonnegative, sublinear and continuous
order-preserving function for a not necessarily complete preorder on a real
convex cone in an arbitrary topological real vector space. As a corollary of
the main result, we present necessary and sufficient conditions for the
existence of such an order-preserving function for a complete preorder.
|
math
|
1,264 |
Some equivalences for Martin's Axiom in asymmetric topology
|
math.GN
|
We find some statements in the language of asymmetric topology and continuous
partial orders which are equivalent to the statements $\kappa < \mathfrak m$ or
$\kappa < \mathfrak p$.
|
math
|
1,265 |
The maximal G-compactifications of G-spaces with special actions
|
math.GN
|
An action on a G-space induces uniformities on the phase space. It is shown
when the maximal G-compactification of a G-space can be obtained as a
completion of the phase space with respect to one of these uniformities.
Structure of G-spaces with special actions is investigated.
|
math
|
1,266 |
van Douwen's problems related to the Bohr topology
|
math.GN
|
We comment van Douwen's problems on the Bohr topology of the abelian groups
raised in his paper (The maximal totally bounded group topology on G and the
biggest minimal G-space for Abelian groups G) as well as the steps in the
solution of some of them. New solutions to two of the resolved problems are
also given.
|
math
|
1,267 |
On Tychonoff-type hypertopologies
|
math.GN
|
In 1975, M. M. Choban introduced a new topology on the set of all closed
subsets of a topological space, similar to the Tychonoff topology but weaker
than it. In 1998, G. Dimov and D. Vakarelov used a generalized version of this
new topology, calling it Tychonoff-type topology. The present paper is devoted
to a detailed study of Tychonoff-type topologies on an arbitrary family M of
subsets of a set X. When M contains all singletons, a description of all
Tychonoff-type topologies O on M is given. The continuous maps of a special
form between spaces of the type (M,O) are described in an isomorphism theorem.
The problem of commutability between hyperspaces and subspaces with respect to
a Tychonoff-type topology} is investigated as well. Some topological properties
of the hyperspaces (M,O) with Tychonoff-type topologies O are briefly
discussed.
|
math
|
1,268 |
Chainable subcontinua
|
math.GN
|
This paper is concerned with conditions under which a metric continuum (a
compact connected metric space) contains a non-degenerate chainable continuum.
|
math
|
1,269 |
On finite T_0 topological spaces
|
math.GN
|
Finite topological spaces became much more essential in topology, with the
development of computer science. The task of this paper is to study and
investigate some properties of such spaces with the existence of an ordered
relation between their minimal neighborhoods. We introduce notations and
elementary facts known as Alexandroff space. The family of minimal
neighborhoods forms a unique minimal base. We consider T_0 spaces. We give a
link between finite $T_0$ spaces and the related partial order. Finally, we
study some properties of multifunctions and their relationships with connected
ordered topological spaces.
|
math
|
1,270 |
Transfinite sequences of continuous and Baire 1 functions on separable metric spaces
|
math.GN
|
We investigate the existence of well-ordered sequences of Baire 1 functions
on separable metric spaces.
|
math
|
1,271 |
Characterizing continuity by preserving compactness and connectedness
|
math.GN
|
Let us call a function $f$ from a space $X$ into a space $Y$ preserving if
the image of every compact subspace of $X$ is compact in $Y$ and the image of
every connected subspace of $X$ is connected in $Y$. By elementary theorems a
continuous function is always preserving. Evelyn R. McMillan proved in 1970
that if $X$ is Hausdorff, locally connected and Frechet, $Y$ is Hausdorff, then
the converse is also true: any preserving function $f:X\to Y$ is continuous.
The main result of this paper is that if $X$ is any product of connected
linearly ordered spaces (e.g. if $X = R^\kappa$) and $f:X \to Y$ is a
preserving function into a regular space $Y$, then $f$ is continuous.
|
math
|
1,272 |
On the metrizability of spaces with a sharp base
|
math.GN
|
A base $\mathcal{B}$ for a space $X$ is said to be sharp if, whenever $x\in
X$ and $(B_n)_{n\in\omega}$ is a sequence of pairwise distinct elements of
$\mathcal{B}$ each containing $x$, the collection $\{\bigcap_{j\le
n}B_j:n\in\omega\}$ is a local base at $x$. We answer questions raised by
Alleche et al. and Arhangel$'$ski\u{\i} et al. by showing that a pseudocompact
Tychonoff space with a sharp base need not be metrizable and that the product
of a space with a sharp base and $[0,1]$ need not have a sharp base. We prove
various metrization theorems and provide a characterization along the lines of
Ponomarev's for point countable bases.
|
math
|
1,273 |
Sub-representation of posets
|
math.GN
|
We define a property sub-representability and we give a complete
characterisation of sub-representability of posets.
|
math
|
1,274 |
Fell-continuous selections and topologically well-orderable spaces II
|
math.GN
|
The present paper improves a result of V. Gutev and T. Nogura (1999) showing
that a space $X$ is topologically well-orderable if and only if there exists a
selection for $\mathcal{F}_2(X)$ which is continuous with respect to the Fell
topology on $\mathcal{F}_2(X)$. In particular, this implies that
$\mathcal{F}(X)$ has a Fell-continuous selection if and only if
$\mathcal{F}_2(X)$ has a Fell-continuous selection.
|
math
|
1,275 |
Special metrics
|
math.GN
|
This is a survey on special metrics. We shall present some results and open
questions on special metrics mainly appeared in the last 10 years
|
math
|
1,276 |
Compactification of a map which is mapped to itself
|
math.GN
|
We prove that if $T: X \to X$ is a selfmap of a set $X$ such that $\bigcap
\{T^{n}X: n\in N}\}$ is a one-point set, then the set $X$ can be endowed with a
compact Hausdorff topology so that $T$ is continuous.
|
math
|
1,277 |
Sequence of dualizations of topological spaces is finite
|
math.GN
|
Problem 540 of J. D. Lawson and M. Mislove in Open Problems in Topology asks
whether the process of taking duals terminate after finitely many steps with
topologies that are duals of each other. The problem for $T_1$ spaces was
already solved by G. E. Strecker in 1966. For certain topologies on hyperspaces
(which are not necessarily $T_1$), the main question was in the positive
answered by Bruce S. Burdick and his solution was presented on The First
Turkish International Conference on Topology in Istanbul in 2000. In this paper
we bring a complete and positive solution of the problem for all topological
spaces. We show that for any topological space $(X,\tau)$ it follows
$\tau^{dd}=\tau^{dddd}$. Further, we classify topological spaces with respect
to the number of generated topologies by the process of taking duals.
|
math
|
1,278 |
On Lebesgue Theorem for multivalued functions of two variables
|
math.GN
|
In the paper we investigate Borel classes of multivalued functions of two
variables. In particular we generalize a result of Marczewski and
Ryll-Nardzewski concerning of real function whose ones of its sections are
right-continuous and other ones are of Borel class $\alpha$, into the case of
multivalued functions.
|
math
|
1,279 |
A survey of J-spaces
|
math.GN
|
This note is a survey of $J$-spaces.
|
math
|
1,280 |
Using nets in Dedekind, monotone, or Scott incomplete ordered fields and definability issues
|
math.GN
|
Given a Dedekind incomplete ordered field, a pair of convergent nets of gaps
which are respectively increasing or decreasing to the same point is used to
obtain a further equivalent criterion for Dedekind completeness of ordered
fields: Every continuous one-to-one function defined on a closed bounded
interval maps interior of that interval to the interior of the image. Next, it
is shown that over all closed bounded intervals in any monotone incomplete
ordered field, there are continuous not uniformly continuous unbounded
functions whose ranges are not closed, and continuous 1-1 functions which map
every interior point to an interior point (of the image) but are not open.
These are achieved using appropriate nets cofinal in gaps or coinitial in their
complements. In our third main theorem, an ordered field is constructed which
has parametrically definable regular gaps but no $\emptyset$-definable
divergent Cauchy functions (while we show that, in either of the two cases
where parameters are or are not allowed, any definable divergent Cauchy
function gives rise to a definable regular gap). Our proof for the mentioned
independence result uses existence of infinite primes in the subring of the
ordered field of generalized power series with rational exponents and real
coefficients consisting of series with no infinitesimal terms, as recently
established by D. Pitteloud.
|
math
|
1,281 |
Proper and admissible topologies in the setting of closure spaces
|
math.GN
|
A Cech closure space $(X,u)$ is a set $X$ with a (Cech) closure operator $u$
which need not be idempotent. Many properties which hold in topological spaces
hold in Cech closure spaces as well. The notions of proper (splitting) and
admissible (jointly continuous) topologies are introduced on the sets of
continuous functions between Cech closure spaces. It is shown that some
well-known results of Arens and Dugundji and of Iliadis and Papadopoulos are
true in this setting. We emphasize that Theorems 1--10 encompass the results of
A. di Concilio and of Georgiou and Papadopoulos for the spaces of
continuous-like functions as $\theta$-continuous, strongly and weakly
$\theta$-continuous, weakly and super-continuous.
|
math
|
1,282 |
Orbits of turning points for maps of finite graphs and inverse limit spaces
|
math.GN
|
In this paper we examine the topology of inverse limit spaces generated by
maps of finite graphs. In particular we explore the way in which the structure
of the orbits of the turning points affects the inverse limit. We show that if
$f$ has finitely many turning points each on a finite orbit then the inverse
limit of $f$ is determined by the number of elements in the $\omega$-limit set
of each turning point. We go on to identify the local structure of the inverse
limit space at the points that correspond to points in the $\omega$-limit set
of $f$ when the turning points of $f$ are not necessarily on a finite orbit.
This leads to a new result regarding inverse limits of maps of the interval.
|
math
|
1,283 |
Wallman-Frink proximities
|
math.GN
|
This is a survey of compactification extension results and problems for a
special class of proximities.
|
math
|
1,284 |
Fuzzy functions and an extension of the category L-Top of Chang-Goguen L-topological spaces
|
math.GN
|
We study FTOP(L), a fuzzy category with fuzzy functions in the role of
morphisms. This category has the same objects as the category L-TOP of
Chang-Goguen L-topological spaces,but an essentially wider class of morphisms -
so called fuzzy functions introduced earlier in our joint work with U. Hohle
and H. Porst.
|
math
|
1,285 |
On lower semicontinuous multifunctions in quasi-uniform and vector spaces
|
math.GN
|
Given a cover $\mathcal{B}$ of a quasi-uniform space $Y$ we introduce a
concept of lower semicontinuity for multifunctions $F:X\to 2^Y$, called
$\mathcal{B}$-lsc. In this way, we get a common description of Vietoris-lsc,
Hausdorff-lsc, and bounded-Hausdorff-lsc as well. Further, we examine
set-theoretical and vector operations on such multifunctions. We also point out
that the convex hull of Hausdorff-lsc multifunctions need not to be
Hausdorff-lsc except the case where the range space is locally convex.
|
math
|
1,286 |
Quasiorders on topological categories
|
math.GN
|
We prove that, for every cardinal number $\alpha\geq {\mathfrak c}$, there
exists a metrizable space $X$ with $|X|=\alpha$ such that for every pair of
quasiorders $\leq_1$, $\leq_2$ on a set $Q$ with $|Q| \leq \alpha$ satisfying
the implication $$q \leq_1 q' \implies q \leq_2 q'$$ there exists a system
$\{X(q) : q\in Q\}$ of non-homeomorphic clopen subsets of $X$ with the
following properties: (1) $q \leq_1 q'$ if and only if $X(q)$ is homeomorphic
to a clopen subset of $X(q')$, (2) $q \leq_2 q'$ implies that $X(q)$ is
homeomorphic to a closed subset of $X(q')$ and (3) $\neg (q \leq_2 q')$ implies
that there is no one-to-one continuous map of $X(q)$ into $X(q')$.
|
math
|
1,287 |
Compactifications of topological groups
|
math.GN
|
Every topological group $G$ has some natural compactifications which can be a
useful tool of studying $G$. We discuss the following constructions: (1) the
greatest ambit $S(G)$ is the compactification corresponding to the algebra of
all right uniformly continuous bounded functions on $G$; (2) the Roelcke
compactification $R(G)$ corresponds to the algebra of functions which are both
left and right uniformly continuous; (3) the weakly almost periodic
compactification $W(G)$ is the envelopping compact semitopological semigroup of
$G$ (`semitopological' means that the multiplication is separately continuous).
The universal minimal compact $G$-space $X=M_G$ is characterized by the
following properties: (1) $X$ has no proper closed $G$-invariant subsets; (2)
for every compact $G$-space $Y$ there exists a $G$-map $X\to Y$. A group $G$ is
extremely amenable, or has the fixed point on compacta property, if $M_G$ is a
singleton. We discuss some results and questions by V. Pestov and E. Glasner on
extremely amenable groups. The Roelcke compactifications were used by M.
Megrelishvili to prove that $W(G)$ can be a singleton. They can be used to
prove that certain groups are minimal. A topological group is minimal if it
does not admit a strictly coarser Hausdorff group topology.
|
math
|
1,288 |
A locally connected continuum without convergent sequences
|
math.GN
|
We answer a question of Juhasz by constructing under CH an example of a
locally connected continuum without nontrivial convergent sequences.
|
math
|
1,289 |
Nonstandard proofs of Eggleston like theorems
|
math.GN
|
We prove theorems of the following form: if $A\subseteq {\mathbb R}^2$ is a
big set, then there exists a big set $P\subseteq {\mathbb R}$ and a perfect set
$Q\subseteq {\mathbb R}$ such that $P\times Q\subseteq A$. We discuss cases
where big set means: set of positive Lebesgue measure, set of full Lebesgue
measure, Baire measurable set of second Baire category and comeagre set. In the
first case (set of positive measure) we obtain the theorem due to Eggleston. In
fact we give a simplified version of the proof given by J. Cichon. To prove
these theorems we use Shoenfield's theorem about absoluteness for
$\Sigma^1_2$-sentences.
|
math
|
1,290 |
Concerning the dual group of a dense subgroup
|
math.GN
|
Throughout this Abstract, $G$ is a topological Abelian group and $\hat{G}$ is
the space of continuous homomorphisms from $G$ into $T$ in the compact-open
topology. A dense subgroup $D$ of $G$ determines $G$ if the (necessarily
continuous) surjective isomorphism $\hat{G} \twoheadrightarrow \hat{D}$ given
by $h\mapsto h|D$ is a homeomorphism, and $G$ is determined if each dense
subgroup of $G$ determines $G$. The principal result in this area, obtained
independently by L. Aussenhofer and M. J. Chasco}, is the following: Every
metrizable group is determined. The authors offer several related results,
including these. (1) There are (many) nonmetrizable, noncompact, determined
groups. (2) If the dense subgroup $D_i$ determines $G_i$ with $G_i$ compact,
then $\oplus_i D_i$ determines $\Pi_i G_i$. In particular, if each $G_i$ is
compact then $\oplus_i G_i$ determines $\Pi_i G_i$. (3) Let $G$ be a locally
bounded group and let $G^+$ denote $G$ with its Bohr topology. Then $G$ is
determined if and only if ${G^+}$ is determined. (4) Let $non(N)$ be the least
cardinal $\kappa$ such that some $X \subseteq T}$ of cardinality $\kappa$ has
positive outer measure. No compact $G$ with $w(G)\geq non(N)$ is determined;
thus if $non(N)=\aleph_1$ (in particular if CH holds), an infinite compact
group $G$ is determined if and only if w(G)=\omega$. Question. Is there in ZFC
a cardinal $\kappa$ such that a compact group $G$ is determined if and only if
$w(G)<\kappa$? Is $\kappa=non(N)$? $\kappa=\aleph_1$?
|
math
|
1,291 |
Proceedings of the Ninth Prague Topological Symposium. Contributed papers from the symposium held in Prague, August 19-25, 2001
|
math.GN
|
This collection of thirty two reviewed articles covers several fields of
General Topology. Several contributions represent invited presentations at the
Ninth Prague Topological Symposium.
|
math
|
1,292 |
Some equivalent of extremally disconnected spaces
|
math.GN
|
We give some equivalent characterizations of exremally disconnected spaces
|
math
|
1,293 |
The Locally Fine Coreflection and Normal Covers in the Products of Partition-complete Spaces
|
math.GN
|
We prove that the countable product of supercomplete spaces having a
countable closed cover consisting of partition-complete subspaces is
supercomplete with respect to its metric-fine coreflection. Thus, countable
products of sigma-partition-complete paracompact spaces are again paracompact.
On the other hand, we show that in arbitrary products of partition-complete
paracompact spaces, all normal covers can be obtained via the locally fine
coreflection of the product of fine uniformities.
|
math
|
1,294 |
Locales, the locally fine construction and formal spaces
|
math.GN
|
We investigate the connection between the spatiality of locale products and
the earlier studies of the author on the locally fine coreflection of the
products of uniform spaces. After giving a historical introduction and
indicating the connection between spatiality and the locally fine construction,
we indicate how the earlier results directly solve the first of the two open
problems announced in the thesis of T. Plewe. Finally, we establish a general
isomorphism between the covering monoids of the localic product of topological
(completely regular) spaces and the locally fine coreflection of the
corresponding product of (fine) uniform spaces. Additionally, paper relates the
recent studies on formal topology and uniform spaces by showing how the
transitivity of covering relations corresponds to the locally fine
construction.
|
math
|
1,295 |
On finite-dimensional maps
|
math.GN
|
Let $f\colon X\to Y$ be a perfect surjective map of metrizable spaces. It is
shown that if $Y$ is a $C$-space (resp., $\dim Y\leq n$ and $\dim f\leq m$),
then the function space $C(X,\uin^{\infty})$ (resp., $C(X,\uin^{2n+1+m})$)
equipped with the source limitation topology contains a dense $G_{\delta}$-set
$\mathcal{H}$ such that $f\times g$ embeds $X$ into $Y\times\uin^{\infty}$
(resp., into $Y\times\uin^{2n+1+m}$) for every $g\in\mathcal{H}$. Some
applications of this result are also given.
|
math
|
1,296 |
On finite-to-one maps
|
math.GN
|
Let $f\colon X\to Y$ be a $\sigma$-perfect $k$-dimensional surjective map of
metrizable spaces such that $\dim Y\leq m$. It is shown that, for every
positive integer $p\geq 1$ there exists a dense $G_{\delta}$-subset ${\mathcal
H}(k,m,p)$ of $C(X,\uin^{k+p})$ with the source limitation topology such that
if $g\in{\mathcal H}(k,m,p)$, then each fiber of $f\triangle g$ contains at
most $\max\{m+k-p+2,1\}$ points.This result provides a proof of two hypotheses
of S. Bogatyi, V. Fedorchuk and J. van Mill.
|
math
|
1,297 |
Partitions of unity
|
math.GN
|
The paper contains an exposition of part of topology using partitions of
unity. The main idea is to create variants of the Tietze Extension Theorem and
use them to derive classical theorems. This idea leads to a new result
generalizing major results on paracompactness (Stone Theorem and Tamano
Theorem), a result which serves as a connection to Ascoli Theorem. A new
calculus of partitions of unity is introduced with applications to dimension
theory and metric simplicial complexes. The geometric interpretation of this
calculus is the barycentric subdivision of simplicial complexes. Also, joins of
partitions of unity are often used; they are an algebraic version of joins of
simplicial complexes.
|
math
|
1,298 |
Approximation of k-dimensional maps
|
math.GN
|
In this paper we prove the equivalence of the questions of B.A. Pasynkov and
V.V. Uspenskij. We also get some partial results answering these questions in
affirmative. As a corollary to these results we get an extention of the
Hurewicz formula to the extensional dimension.
|
math
|
1,299 |
Extraordinary dimension theories generated by complexes
|
math.GN
|
We study the extraordinary dimension function dim_{L} introduced by
\v{S}\v{c}epin. An axiomatic characterization of this dimension function is
obtained. We also introduce inductive dimensions ind_{L} and Ind_{L} and prove
that for separable metrizable spaces all three coincide. Several results such
as characterization of dim_{L} in terms of partitions and in terms of mappings
into $n$-dimensional cubes are presented. We also prove the converse of the
Dranishnikov-Uspenskij theorem on dimension-raising maps.
|
math
|
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