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

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1answer
256 views

Quotient Metric Space [closed]

Let $(X,d)$ be a metric space and $R\subseteq X \times X$ an equivalence relation. There is a condition for which $X/R$ with the usual quotient topology is a metric space? Thanks!
1
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0answers
54 views

Collapsing a countable collection of intervals on $\mathbb{S}^1$

Consider a countable collection $I_n$ of closed connected disjoint intervals on $\mathbb{S}^1$. When this collection is maximal, the set $\bigcap \nolimits_{i=1}^{n}( \mathbb{S}^1 \backslash \bigcup ...
5
votes
1answer
381 views

sets without perfect subset in a non-separable completely metrizable space

Suppose $X$ is a completely metrizable (but not separable) space. Suppose $D$ is a Borel (actually $F_{\sigma}$) subset of $X$. Is there any logical relation between the following statements? [1] ...
6
votes
1answer
388 views

Is there a suitably generalized Baire property for topological spaces of arbitrary cardinalities?

Is there some suitable generalization to the notion of Baire property for topological spaces of arbitrary cardinalities which satisfies the following condition: The meager sets are sets which are ...
20
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1answer
787 views

A function composed with itself produces the identity

Let $B$ be the closed unit ball in $\mathbb R^3$ and $f: B\to B$ continuous, such that $f\circ f$ is the identity (i.e., $f\circ f=\mathbb 1_B$) and $f$ restricted on $\partial B$ is also the identity ...
3
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1answer
115 views

The finest countably generated free topological group so that $x^{m_{n}}_n\rightarrow 1$?

What is the finest free topological group $H$ with generators ${x_{1},x_{2},...}$ so that $x^{m_{n}}_n\rightarrow 1$ for all sequences $m_{1},m_{2},...$? Is $H \simeq K$, with $K$ the natural ...
2
votes
2answers
329 views

Locally compact space that is not topologically complete

It is know that for a metric space, it is locally compact and separable iff exist an equivalent metric where a set is compact iff it is closed and limited. So, locally compact and seperable metric ...
1
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1answer
175 views

algebra-geometry duality

For topological spaces $S$ and $T$, denote by $C(S)$ and $C(T)$ the corresponding algebras of continuous real-valued functions. What are the necessary conditions that we need to impose on $S$ and $T$ ...
4
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0answers
206 views

Is the Milnor construction contractible

Let $G$ be any topological group. Then we can form the infinite join $E_G$ of $G$, i.e. the colimit $G*G*G\cdots$. Is $E_G$ contractible? I mean it is clear that $E_G$ is weakly contractible, ...
0
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0answers
49 views

Minimal length of connection

I wonder if someone has an example which answers the following problem. Find $X\subseteq\mathbb R^n$ bounded, such that the set $$ \{ \mathcal H^1(S) \colon S\cup X \text{ connected}\} $$ does not ...
3
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0answers
224 views

How many ways do we have to prove that a mapping is open?

Given a continuous mapping $f$ between Euclidean domains (or domains in topological manifolds) of the same (topological) dimension, what are the natural assumptions to conclude that $f$ is open? ...
5
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1answer
181 views

Completely Metrizable Space and Baire Theorem

Is well know that completely metrizable spaces are Baire's spaces. Reciprocally, if $X$ is a Baire's metric space, then $X$ is completely metrizable?
2
votes
1answer
196 views

Probability measures on $L^p$

Let $(X,\mathcal X,\mu)$ be a fixed measure space, and suppose that $\mu$ is stationary and ergodic with respect to the (left) action of a topological group $G$. Stationarity means that $\mu = g_* \mu ...
20
votes
1answer
530 views

Which spaces have the (weak) homotopy type of compact Hausdorff spaces?

Inspired by the discussion in the comments of this question, I'd like to ask the following question: is it possible to characterize the class of spaces that are homotopy equivalent (or weak ...
2
votes
1answer
143 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
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1answer
347 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
125 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 ...
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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
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0answers
143 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
135 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
224 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
200 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
222 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
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1answer
158 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
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1answer
110 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 ...
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2answers
732 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
172 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
151 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
198 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
212 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
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0answers
143 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 $f: S\to S$ ...
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3answers
182 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
75 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
195 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
224 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
799 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
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0answers
113 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
88 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
58 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?
13
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4answers
486 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
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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
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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
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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
117 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
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1answer
280 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
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1answer
89 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
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1answer
230 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
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0answers
139 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
368 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 ...
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0answers
232 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 ...