**1**

vote

**0**answers

24 views

### Number of faces of a polytopal subdivision

Let $\mathcal{P}$ be a (bounded) polytope in $\mathbb{R}^d$ and let $\mathcal{C}$ be a polytopal subdivision of $\mathcal{P}$ [1].
Is there a known tight upper bound in the number of polytopes in ...

**-1**

votes

**0**answers

33 views

### Closed sets on the product space and operators [on hold]

$H$ is an hilbert space and $C$ is a closed subset of $H\times H$ with the product topology. If $P$ is the projection $P: (x,y) \in F\times F \to y \in F$ do we have that
the set $$P(C)= \{ P((x,y)), ...

**1**

vote

**0**answers

33 views

### d-refining covering of normal space

If $X$ is normal, it is well known that for any open-covering $(U_i)$ of $X$, there exist closed subspaces $F_i$ and $G_i$ and an open subspaces $O_i$ such that $$F_i\subset O_i\subset G_i\subset ...

**-1**

votes

**0**answers

43 views

### Open Course or Video lecture for topology [on hold]

i want to take courses of general topology.
so is there any video lecture or open courses of introduction to topology?
and if i want to study modern topology, first of all, it is essential to learn ...

**-4**

votes

**0**answers

63 views

### Non connected topological space with intermediate value Theorem [on hold]

Does there exist a topological space X which is not connected but satisfy intermediate value theorem(IVT).
Where IVT sates: if f is continuous function from X to Y where Y is ordered set in order ...

**3**

votes

**1**answer

174 views

### Quotients of standard Borel spaces

Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation ...

**15**

votes

**2**answers

470 views

### Topological transversality

Warmup question:
Let us say that two continuous functions $f,g:[0,1]\to \mathbb R$ are topologically transverse if their difference $f-g$ has only finitely many zeros, and each zero separates an ...

**2**

votes

**0**answers

47 views

### Separation of topological group elements by invariant neighbourhooods

Let $G$ be a topological group that is Hausdorff, that is, for every pair $(g,h)$ of distinct elements of $G$, there exist disjoint open sets $U_g$ and $U_h$ such that $g \in U_g$ and $h \in U_h$.
...

**4**

votes

**2**answers

111 views

### Are Hausdorff compactifications of a Tychonoff space $X$ in one-to-one correspondence with completely regular subalgebras of $BC(X)$?

Let $X$ be a completely regular (Tychonoff) topological space. It is known that if $\mathscr F\subseteq C(X,[0,1])$ separates points and closed sets (that is, for every closed set $E\subseteq X$ and ...

**4**

votes

**1**answer

90 views

### Is an extension of compact Hausdorff topological groups compact?

Let $1 \rightarrow A \xrightarrow{a} B \xrightarrow{c} C \rightarrow 1$ be a short exact sequence of topological groups (i.e., all maps are continuous, $A = \mathrm{Ker}(c)$, and $C = ...

**16**

votes

**2**answers

492 views

### Which sets occur as boundaries of other sets in topological spaces?

This question was originally asked on MathStackExchange and is migrated here with opinion from MO meta. I am integrating the inputs from users Daniel Fischer and Emil Jerabek there into this post.
...

**5**

votes

**1**answer

131 views

### A realcompact analogue of the Baire category theorem

Let $\frak{m}$ be the least measurable cardinal. A space $X$ is realcompact if it is homeomorphic to a closed subset of some product $\mathbb{R}^I$. Let $X$ be realcompact with $P_\frak{m}$ topology, ...

**9**

votes

**2**answers

181 views

### A differentiable one-parameter family of codimension 2 subspaces of $\mathbb{C}^n$ cannot fill $\mathbb{C}^n$, right?

Suppose that $P(t)$ is a one-parameter family of rank 2 self-adjoint projections on $\mathbb{C}^n$ that vary analytically in the real parameter $t \in [0,1]$. I claim that there must exist a vector $x ...

**12**

votes

**1**answer

203 views

### A generalization of the Arhangelskii Theorem

Arhangeleskii's Theorem states the following
For any Hausdorff topological space $X$,
$$
|X|\leq2^{\chi(X)L(X)}
$$
where $\chi(X)$ is the character of $X$ and $L(X)$ is the Lindelöf degree of ...

**0**

votes

**0**answers

54 views

### Can a “weak” topological space be a Moore space?

Let B be a reflexive and infinite dimensional real Banach space-which could be Hilbert space l^2- and let B be endowed with the weak topology. Although this topology is regular and Hausdorff, it is ...

**3**

votes

**3**answers

352 views

### “countable” topology

Given universal set $U$. Is there any name of the collection of subsets of $U$ (call them quasi-open) satisfying the following axioms:
i) $\emptyset$ and $U$ are quasi-open;
ii) finite intersections ...

**2**

votes

**0**answers

31 views

### Relative isotopy of simple curves in a disk

Consider the closed two dimensional disk and two fixed points $A$ and $B$ on the boundary. I am looking for a reference for the following fact: a simple topological curve with end points $A$ and $B$ ...

**1**

vote

**0**answers

120 views

### Homeomorphism of compact Hausdorff spaces

(Note: I asked this question at MSE over a day ago and received no answer, so I'm now reposting it here. Link: http://math.stackexchange.com/questions/853500/homeomorphism-of-compact-hausdorff-spaces)
...

**2**

votes

**0**answers

72 views

### Berkovich Analytification of the transseries

I am looking for references to articles about the following subjects:
Connections from the field of (real) transseries to the field of surreal numbers (mentioned very briefly in the introduction of ...

**6**

votes

**1**answer

184 views

### Homeomorphism between derived sets implies homeomorphism

(Note: I asked this question at MSE days ago and received no answer, so I'm now reposting it here.)
I want to prove the following statement:
Let $K_1$ and $K_2$ be two countable, compact sets of ...

**2**

votes

**1**answer

91 views

### open subsets of boundary [closed]

Let $\bar{X}$ be a Hausdorff and compact topological space. Suppose that $X$ is an open and dense subset of $\bar{X}$. Let $\nu X=\bar{X}\setminus X$ and assume that $U\subseteq \nu X$ is an open ...

**3**

votes

**0**answers

127 views

### Compact set covered by two opens

The following lemma about locally compact (but not necessarily Hausdorff) spaces or continuous lattices appears frequently but without citation. It is easy to prove but important in proofs.
If a ...

**3**

votes

**1**answer

145 views

### A question on $Z^{*}$ algebras

A $Z^{*}$ algebra is a $C^{*}$ algebra which satisfies each of the following equivalent conditions:
All elements of $A$ are left zero divisor.
All elements are right zero divisor.
All elements ...

**2**

votes

**2**answers

179 views

### Compact, densely ordered spaces

During my work with order preserving homeomorphisms, I got interested in the double arrow space and, subsequently, in the lexicographic square.
I would really like to find examples of spaces like ...

**1**

vote

**1**answer

147 views

### Sober topological subspace

Assume $X$ to be a Notherian topological space such that any irreducible closed
subset has a unique generic point. Consider $Y\subseteq X$ as a topological space with the induced topology from $X$. Is ...

**3**

votes

**2**answers

269 views

### If $E$ maps onto a contractible space with contractible fibers, must $E$ be contractible?

Let $p\colon E\to C$ be a continuous, surjective map between topological spaces with $C$ contractible. Suppose that $p^{-1}(c)$ is contractible for each $c\in C$. Is it true that $E$ is weakly ...

**1**

vote

**1**answer

62 views

### Unitization via “End points compactification”

We know that every compactification of locally compact Hausdorff spaces correspond to a unitization of $C^{*}$ algebras. For example the one point compactification corresponds to the minimal ...

**3**

votes

**1**answer

208 views

### Closed Graph Theorem and Spaces Of Continuous Functions

Let $X$ be a (Tychonoff) topological space. Consider $C\left(X\right)$ being a topological vector space of all continuous scalar-valued functions with the compact-open topology.
Assume that $Y$ is a ...

**0**

votes

**1**answer

224 views

### Heisenberg group acts on the circle

Let $H$ be a Heisenberg group, i.e.
$$
H=\left\langle a,b,c |[a,b]=c,[a,c]=[b,c]=1\right\rangle.
$$
$H$ acts on the circle by homeomorphism which preserves the orientation. If the rotation number of ...

**5**

votes

**0**answers

182 views

### Homeomorphisms of product spaces: an example

In the first of these lectures (http://www.mpim-bonn.mpg.de/node/4436) given by M. Freedman he says that there exists (compact metric) spaces $X$ and $Y$ such that $X\times S^{1}$ is homeomorphic to ...

**8**

votes

**1**answer

408 views

### Why does the singular simplicial space geometrically realize to the original space?

I have seen it claimed that (for compactly generated Hausdorff spaces) the geometric realization of the singular (internal) simplicial space is homotopy equivalent to the original space. I know how to ...

**8**

votes

**1**answer

250 views

### Is $\ell^\infty$ Polishable?

Consider $\ell^\infty$ as a subspace of the Polish space $\mathbb{R}^\omega$. It is easy to check that $\ell^\infty$ is not Polish in the subspace topology, as it is countable union of the compact ...

**5**

votes

**3**answers

158 views

### Sequential closure of a set: standard terminology, notation, and properties

Let $X$ be a topological vector space (or, perhaps, more generally uniform space). Let $A\subset X$ be a subset. Let $A^s$ denote the set of limits of all convergent sequences (I guess $A^s$ is called ...

**7**

votes

**1**answer

199 views

### C* algebras of Almost Periodic Functions

Suppose we take, for example, the $C^*$-algebra which is the sup norm closure of the exponentials $e^{2 \pi i ax}$ where $a \in \mathbb{Z} + \theta \mathbb{Z}$ for $\theta$ an irrational number. This ...

**7**

votes

**0**answers

119 views

### How much of general topology can be developed by taking the notion of “connected set” as the sole topological primitive

Let X be an infinite regular topological space which is connected and locally connected. If no point of X is a cut point, does X always have base of connected open sets whose complements (with respect ...

**3**

votes

**1**answer

95 views

### Ultrafilters of weight $\aleph_2$ in Sacks model

It is well-known that in Sacks model there are P-points and even Ramsey ultrafilters, but what the usual (i.e. findable in the literature) proofs for these facts do is proving that ground model ...

**24**

votes

**1**answer

684 views

### Which powers of the closed unit interval are homeomorphic?

It is known that no two distinct finite powers of the closed unit interval are homeomorphic:
$I^m$ is homeomorphic to $I^n$ iff $m=n$. (Brouwer, Lebesgue, 1911)
Is the analogous result for infinite ...

**4**

votes

**2**answers

177 views

### Question about lower homology class of cobordism

Assume there are three differential oriented manifold $M_0$, $M_1$, $W$ with $\partial W= M_0 \coprod -M_1$. Denote dim $M_0$=dim $M_1$=n, and dim W=n+1.
We know that for the highest homology class, ...

**2**

votes

**0**answers

132 views

### The mathematics in understand anyons [closed]

I've been about particles called anyons which exist within a two dimensional framework. I've also found out that these particles can have an angular momentum equal to any real number. Normally, in ...

**0**

votes

**0**answers

89 views

### topological space of Wang Tile

When trying to reprove a theorem in Wang tile:
An established proof in Wang Tile which I doubt
, a few notions are provided which I would like to seek for more information:
For a given set of blocks ...

**5**

votes

**0**answers

83 views

### Tensor product of dual groups

Let $G,H$ be compact abelian groups, $G^*,H^*$ be their Pontryagin duals, $G^*\otimes H^*$ the tensor product of $G^*,H^*$ and $K=(G^*\otimes H^*)^*$. Does the group $K$ have a special name? What is ...

**2**

votes

**0**answers

294 views

### Topological proof of a result in Logic

I proved the result below using logic. My questions:
Can this theorem be proved by purely topological means?
Do you know any theorems that either can be used to prove the same result, or which give ...

**9**

votes

**1**answer

285 views

### continuous images of open intervals

The well-known Hahn-Mazurkiewicz theorem characterizes those nonempty Hausdorff spaces $X$ that admit a continuous surjection $\alpha: [0, 1] \to X$ from the closed unit interval: it is necessary and ...

**5**

votes

**2**answers

291 views

### Beyond Cantor's Teepee

From Counterexamples in Topology by Steen and Seebach (2nd edition) example 129 page 145 we have an example of connected and totally path-disconnected space.
It is defined as follow:
Fix $p= ...

**1**

vote

**1**answer

79 views

### Properties of open covers

I am reading this article in which two properties of open covers are described:
$\gamma$-property: If $\mathcal U$ is an open $\omega$-cover of $X$, then there sequence $\{ G_n : G_n \in \mathcal ...

**3**

votes

**1**answer

263 views

### Realizing homomorphisms between fundamental groups

Let $X,Y$ be compact connected manifolds and $\varphi\colon\pi_1(X)\to\pi_1(Y)$ be a homomorphism between their fundamental groups. Under what conditions on $X$, $Y$ and $\varphi$ is it true that ...

**4**

votes

**1**answer

311 views

### Existence of infinite groups that are too reluctant to be topological

With ZFC, is there an infinite group $G$ such that there is no non-trivial non-discrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{-1}$ ...

**0**

votes

**3**answers

171 views

### Lifts across covering maps

Let $X,Y,Z$ be connected topological spaces, $f\colon X\to Y$ be a continuous map and $p\colon Z\to Y$ be a covering map. The problem is the existence of a continuous lift of $f$ across $p$. A ...

**0**

votes

**1**answer

584 views

### What does the 3rd axiom of topologies defined by neighbourhood mean? [closed]

Recall the axioms of a topology defined in terms of neighbourhoods, we call a topology on $X$ a family $(\mathcal{V}_x)_{x\in X}$ of sets in $\mathcal{P}(\mathcal{P}(X))$ which verifies for all $x\in ...

**15**

votes

**5**answers

1k views

### A topological concept dual to compactness

We say that a subset A in a topological space X is anti-compact if every covering of A by closed sets has a finite subcover. Clearly if X is Hausdorff then all anti-compact subsets of X are finite. ...