Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

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2
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1answer
139 views

Name of the concept “Topological boundary of A intersected with A”

In closure spaces (thus, also in topological spaces), one may define the boundary of a set A as the closure of A minus the interior of A. This set is partitioned into "the closure of A minus A" and "A ...
5
votes
1answer
333 views

Spaces over which every vector bundle is a summand of the trivial bundle

Let X be a Hausdorff space such that every real vector bundle on X is summand of a trivial bundle. Does this imply that X is homotopy equivalent to a compact Hausdorf space? This question ...
2
votes
0answers
122 views

How many ways we have to prove that a topologically (or analytically) nice mapping is injective?

I would like to know what are the methods people have used to prove that a topologically (or analytically) nice mapping $f: B\to \Omega$ is injective? Above, $B$ is the unit ball in $\Bbb R^n$ and ...
1
vote
0answers
92 views

Characterising singular homology among a more general class of cosimplicial spaces

Is there a way to characterise (up to isomorphism) the cosimplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...
2
votes
0answers
132 views

Comparing two metrics on the space of infinite sequences and relating open and closed sets

Let $X = \{ 0, 1 \}$ and $X^{\mathbb N_0} = \{ x_0 x_1 x_2 \ldots : x_i \in X \}$ be the space of all infinite sequences, then a metric could be defined on it $$ d(u,v) := \frac{1}{2^r} \mbox{ with } ...
2
votes
1answer
134 views

When is a space of probability measures not perfectly normal?

I am looking for examples of pairs ($(\Omega,\Sigma)$, ($\mathcal P(\Omega)$, $\tau$)), where $(\Omega,\Sigma)$ is a measurable space and ($\mathcal P(\Omega)$, $\tau$) is a space of probability ...
2
votes
1answer
215 views

topological group that is connected and locally connected but not path-connected

Is there a ($\mathrm{T}_0$) topological group that is connected and locally connected but is not path-connected? This is a cross-post from MSE, since my question there was posted over three weeks ...
4
votes
1answer
193 views

Hausdorff metric on C[0,1]

Let us consider $C[0,1]$, the space of continuous functions $f\colon [0,1] \to \mathbb{R}$. It comes usually with the metric of the maximum, or of the supremum, $d_{L^{\infty}}$. Each element $f$ in ...
5
votes
2answers
219 views

“Abnormal” manifold

We often assume manifolds to be paracompact Hausdorff. Clearly, this implies normal. However, there is a manifold (I mean locally Euclidean Hausdorff space) which is not paracompact. Without ...
6
votes
1answer
155 views

Knots indistinguishable by HOMFLY

Is there any list (incomplete of course) of knots, that have similar HOMFLY polynomials? I am mainly interested in torus knots.
1
vote
1answer
100 views

Relative interior and dense subsets

(This is a cross-post from here.) Let $A,B\subseteq \mathbb R^d$ be non-empty, such that $B\subseteq \overline A.$ For $S\subseteq\mathbb R^d$ define the relative interior of $S$ by ...
16
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2answers
717 views

An order type $\tau$ equal to its power $\tau^n, n>2$

(This is a re-post of my old unanswered question from Math.SE) For purposes of this question, let's concern ourselves only with linear (but not necessarily well-founded) order types. Recall that: ...
4
votes
1answer
171 views

Contractibility of connected holomorphic dynamics?

Let $f$ be a function, holomorphic in $\mathbb{C}$, and $K(f)$ its non-escaping set : $$K(f) = \{ z \in \mathbb{C} : f^{(k)}(z) \nrightarrow_{k \to \infty} \infty \} $$ Question : If $K(f)$ ...
4
votes
1answer
145 views

Statistical models in terms of families of random variables

A statistical model is a function $P : \Theta \to \Delta(X)$, where $\Theta$ is a parameter space, and $\Delta(X)$ is the set of probability measures on a state space $X$. Suppose that $\Theta$ and ...
5
votes
1answer
176 views

Nonmetrizable compact totally disconnected spaces without isolated points

Brouwer proved that a topological space is homeomorphic to the Cantor set if and only if 1) it's non-empty, 2) it's compact, 3) it's totally disconnected, 4) it has no isolated points, and 5) ...
3
votes
1answer
206 views

Continuous Functions

Is there a pair of continuous surjective functions say $f_1$ and $f_2$ from $\mathbb{Q}$ to itself such that for every $x, y \in \mathbb{Q}$, $f_1^{-1}(x) \cap f_2^{-1}(y)$ is non-empty?
1
vote
0answers
135 views

The image of homomorphism of fundamental group of closed surface [closed]

$\phi: \pi_1(S)\to \pi_1(S)$ is a homomorphism of fundamental group of closed orientable surface $S$ of genus >=2. If $\phi$ is not an epimorphism, can we find a non-surjective self map $f: S\to S$ ...
1
vote
3answers
173 views

Axiomatization of locally compact Hausdorff spaces via compact subspaces

The usual axiomatization of a topological space (in the sense of Bourbaki) goes by declaring certain subsets as being open and such that a few axioms are fulfilled by the family of open subsets. It ...
1
vote
1answer
74 views

Simply connectedness of minimal resolution of Kleinian singularities

Is the minimal resolution of Kleinian singularities of type $D_k$ (i.e. the minimal resolution of singularities of the action of the binary dihedral group of order $4(k-2)$ on $ C^2$ simply connected? ...
5
votes
1answer
189 views

If a subset and its complement are path-connected, an neighborhood of the subset is path-connected

I apologize if this is too elementary for this site. Given a closed subset, $X\subset \mathbb{R}^n$, given $X$, $X^C$ path-connected, show that any path-connected neighborhood of $X$, denoted $M$, ...
1
vote
1answer
218 views

(n-1)-dimensional normal currents and Smirnov's paper

I don't know much about currents, but I saw a paper of Smirnov which seems relevant to a problem I am working on. In the very last paragraph of page 848 of the following paper ...
7
votes
3answers
787 views

Is the list of “known” 3D compact manifolds complete?

"it is an open question if the known compact manifolds in 3-D are complete." This is a quote from Eric Weisstein's CRC Concise Encyclopedia of Mathematics, Second Edition. 2010, p.480. (Google ...
4
votes
0answers
111 views

Pettis Integrability and Laws of Large Numbers

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $V$ be a topological vector space with a dual space that separates points. Let $v_n : \Omega \to V$ be a sequence of Pettis ...
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0answers
86 views

non-Hausdorff paratopological group with closed intersection of neighborhoods of 1

Do you have an example of a non-Hausdorff (preferably locally compact) paratopological group $(G,\mathcal T)$ in which the intersection of all neighborhoods of a point is closed?
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0answers
56 views

Is there a collectionwise normal topological vector space which is not paracompact?

I am looking for an example of a collectionwise normal topological vector space that is not paracompact. Any idea about it?
12
votes
4answers
473 views

Continuity on a measure one set versus measure one set of points of continuity

In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$? Now more carefully, with some notation: Suppose $(X, d_X)$ ...
0
votes
0answers
58 views

How to construct point finite covering in collectionwise normal spaces

I am actually looking for a related reference (and ideally if anyone knows the answer) on the following construction problem: Let X= $\prod_{i=1,..,n} X_{i}$ be a collectionwise normal and Hausdorff ...
27
votes
2answers
2k views

Dynamical properties of injective continuous functions on $\mathbb{R}^d$

Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function. Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e. $\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all ...
0
votes
0answers
43 views

tensors invariant under irregular flow

Suppose $M$ is a compact smooth manifold, and let $R$ be a nowhere-vanishing vector field. Then in the irregular scenario, the closure of $R$-orbits is a $r$-torus $T^r$. Now suppose there is a ...
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0answers
112 views

Subdividing Simplicial complexes

Barycentric subdivision is an important tool in simplicial homology theory, where it is used as a means of obtaining finer simplicial complexes from a given one. Are there other common subdivision ...
2
votes
1answer
275 views

Is this a closed set?

Let $\Theta$ and $X$ be two (Hausdorff) topological spaces. Let $\mathbb P : \Theta \to \Delta(X)$ be a "statistical model", i.e., a continuous function from parameter space $\Theta$ to the space of ...
1
vote
1answer
87 views

Banach Isomomorphic Cts Fucntion Algebras for two Non-Homeomorphic Top Spaces?

Can anyone provide me with an example of two non-homeomorphic locally-compact Hausdorff spaces $X$ and $Y$, such that $C(X)$ and $C(Y)$ are isomorphic as Banach algebras. Clearly, the ...
2
votes
1answer
218 views

Dual of the space of continuous functions

Let $T \subseteq \mathbb R$ be a closed set of real numbers. Let $X := C(T, \mathbb R)$ denote the Fréchet space of continuous real-valued functions on $T$. The topology on $X$ is generated by ...
5
votes
0answers
137 views

Series in topological rings that only converge if almost all summands are zero

While trying to understand a certain topological ring better, I stumbled onto the following question. Suppose $I$ is a fixed infinite index set, $R$ is a topological ring and $(x_i)_{i\in I}$ is a ...
4
votes
1answer
326 views

Topological Problems Solved by Lattice Duality

It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ...
12
votes
0answers
220 views

Connected and locally connected, but not path-connected

Allow me to use some non-standard terminology: A h-contractible space is a non-empty topological space $X$ such that, for any topological space $T$ and any pair of continuous maps $f_0, f_1 : T \to ...
3
votes
2answers
164 views

Product of Topological Measure Spaces

Def. A Radon measure $\mu$ on a compact Hausdorff space $X$ is uniformly regular if there is a countable family $\mathcal{A}$ of compact $G_\delta$-subsets of $X$ such that for every open set ...
1
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0answers
131 views

Extending a homeomorphism from a dense set [closed]

Let $X$ and $Y$ be Hausdorff topological spaces, and let $f : X \to Y$ be a Borel-measurable function. Suppose that $D \subseteq X$ is dense, that the image $f(D) \subseteq Y$ is dense, and that $f$ ...
4
votes
4answers
619 views

Categorical Construction of Quotient Topology?

The product topology is the categorical product, and the disjoint union topology is the categorical coproduct. But the arrows in the characteristic diagrams for the subspace and quotient topologies ...
12
votes
2answers
250 views

If G is a sequential topological group, must G x G be sequential?

Using standard definitions, the topological space $Y$ is sequential if for each nonclosed $A \subset Y$, there exists a convergent sequence $a_{1}$ , $a_{2}$,...$\rightarrow b$ so that $a_{n} \in A$ ...
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0answers
136 views

Can a compact metrizable space be determined by its Hausdorff measures?

Suppose that $(X,d)$ is a compact metric space. Now suppose that $h:[0,a]\rightarrow[0,b]$ is a continuous function with $h(0)=0$ where if $x\leq y$, then $h(x)\leq h(y)$. Then define ...
0
votes
0answers
85 views

Analytic extension from a closed analytic subset

I have the following question: Let $\Omega \subset \mathbb{R}^{n}$ be an open set and consider $X \subset \Omega$ an analytic subset. By this I mean that there exists analytic functions ...
2
votes
2answers
184 views

Tychonoff spaces and ideals

Let $X$ be a tychonoff space and let $T$ be the set of all $f \in C(X)$ such that for any $g$ the equation $fg = 1$ has at most finitely many solutions. Under what conditions on $X$, the set $T$ is an ...
5
votes
2answers
456 views

CW complex and group action

This is a general question and any reference or related result will be extremely helpful. Suppose $X$ is a Hausdorff topological space. Suppose G (a countable group) acts on it. Let $Y=X/G$ be the ...
2
votes
2answers
197 views

A Fixed point Theorem that does not need the convexity of set valued map?

I am looking for a fixed point theorem for set valued maps that does not assume the set valued map should be convex valued. Something like contractiblity or other properties can be replaced with ...
3
votes
1answer
151 views

Extending the hyperbolic splitting on $\Lambda$ to a neighborhood of $\Lambda$

Let $M$ be a compact Riemannian manifold and let $f:M→M$ a diffeomorphism. Let $\Lambda\subset M$ be a compact invariant subset of $M$. We say that $\Lambda $ is a hyperbolic set for $f$ when there ...
2
votes
2answers
163 views

A set intersecting the graph of any continuous function in a finite set

New version of the problem I am looking for a characterization of those completely regular and hausdorff spaces $X$ such that the follwing is true: If $f :X\longrightarrow \Bbb{R}$ is continuous ...
1
vote
1answer
574 views

Topological razors (ball-like spaces)

Introduction Many admire the Euclidean space, and I am not an exception. I will try to catch the topological roundness of the $n$-ball in its greatest generality. I call the resulting axiomatized ...
2
votes
1answer
132 views

Is there a maximal (or maximal Tychonoff) non normal space?

Is there a maximal (or maximal Tychonoff) non normal space? In "A Problem of Set-Teoretic Topology" the existence of a maximal Tychonoff space is asserted. Also there exists a perfectly normal maximal ...
16
votes
2answers
492 views

Intersection of compact sets in the unit interval

Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr ...