**5**

votes

**1**answer

149 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

197 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

263 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

62 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

388 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

42 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 $B$ ...

**2**

votes

**0**answers

145 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

89 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

225 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

99 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

151 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 ...

**4**

votes

**1**answer

191 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

**3**answers

240 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

167 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 ...

**4**

votes

**2**answers

292 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

77 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

**0**answers

234 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

247 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

223 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 ...

**9**

votes

**1**answer

465 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 ...

**9**

votes

**1**answer

292 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

212 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

230 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

149 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

103 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

749 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

193 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

145 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 ...

**5**

votes

**0**answers

93 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 ...

**3**

votes

**0**answers

325 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 ...

**10**

votes

**1**answer

345 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

312 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

89 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

322 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 ...

**7**

votes

**1**answer

388 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

189 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

609 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 ...

**16**

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. ...

**1**

vote

**1**answer

150 views

### Modern reference request concerning Efimov's “On dyadic spaces”

Is there any modern reference (book, textbook, monograph, etc.) that contains the following result of B. Efimov (On dyadic spaces // Dokl. Akad. Nauk SSSR 151 (1963) (Russian). English translation: ...

**1**

vote

**0**answers

86 views

### Regarding graphs of continuous functions between zero dimensional spaces

Background for the question: Let for any topological space $B$, $I(B)$ denote the topological space which has the same set of points as of $B$, and the topology is generated by closed and open sets of ...

**5**

votes

**1**answer

194 views

### Example: a locally convex TVS which is not compactly generated

Is there an example of a locally convex topological vector space which is not compactly generated?
(any such example must be non-Fréchet, since all Fréchet spaces are compactly generated)
(note: I ...

**0**

votes

**0**answers

78 views

### Is currying a homeomorphism of array spaces?

Define an "index space" to be any topological space $I$, which is separable and locally compact Hausdorff. Define a "value space" to be any topological vector space $V$, which is separable, locally ...

**0**

votes

**0**answers

166 views

### A noncommutative analogy of the tube lemma

Assume that $A$ and $B$ are two unital commutative Banach algebras. Assume that $\phi \in \mathcal{M} (A)$ is an element of the maximal Ideal space. Define $\alpha: A\hat{\otimes} B \to ...

**4**

votes

**0**answers

187 views

### Intersections of open sets and $\alpha$-favorable spaces

I would like to ask about some classes of topological spaces whether they have been studied, what are they called (if they have a name) and whether they have some interesting properties. For the ...

**3**

votes

**1**answer

274 views

### Suslin lines hereditarily Lindelof

I need to prove that every suslin line is hereditarily Lindelof. Any idea will be helpfull.

**1**

vote

**0**answers

85 views

### First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric

This is a follow up of Question 163246. For the reader's convenience, let me first copy&paste some basic definitions.
We let a semimetric on a set $X$ be a function $d: X \times X \to [0,\infty]$ ...

**18**

votes

**0**answers

581 views

### The topologies for which a presheaf is a sheaf?

Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal.
Suppose that $Q$ is a presheaf on ...

**2**

votes

**2**answers

180 views

### If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?

Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post).
Basically following some ideas of W. Lawvere (but not his ...

**5**

votes

**1**answer

110 views

### When does topological homogeneity imply algebraic homogeneity? Pseudo-arc and Hilbert cube

Knaster's pseudo-arc and Hilbert cube are topologically homogeneous continua. The easier question is: do these spaces admit a topological group structure? (I am sure that the answer is negative). Thus ...

**2**

votes

**3**answers

208 views

### Diffeomorphism with prescribed behaviour

If $\gamma$ and $\eta$ are two smooth curves in a smooth manifold $M$, is it possible to find a diffeomorphism of $M$ such that $f \circ \gamma = \eta$? What if one removes the assumption of ...