**1**

vote

**1**answer

53 views

### Zero-dimensional spaces and clopen separations

Let $X$ be a topological space. (All of the spaces I'm considering are $T_0$, but in general they are not $T_1$. To be even more concrete, one can even consider $X={\rm Spec}(R)$ to be the space of ...

**-4**

votes

**0**answers

36 views

### Are mappings f(a, x) and f(b, x)-c conjugate for some parameter values a, b, and c?

More precisely, what assumptions on f must hold for the above maps to be conjugate?
We know that functions $f$ and $g$ are topologically conjugate, if there exists a homeomorphism $h$ such that ...

**0**

votes

**1**answer

104 views

### Union of connected sets

$\forall \beta \in I$, $A_{\beta }$ is connected, and $\left ( \bigcup_{\alpha < \beta }A_{\alpha } \right )\cap A_{\beta }\neq \varnothing$ . Is $\bigcup_{\alpha \in I}A_{\alpha } $connected?
For ...

**0**

votes

**1**answer

73 views

### Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [on hold]

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology.
Let $X$ be a topological space (for convenience, it might be Polish ...

**1**

vote

**0**answers

48 views

### Digital topology, animal problem, 2-sphere and torus

I have the following question relating digital topology, surfaces, particularly $S^2$ and torus.
Can a body $B$ constructed with cubes (without cavities or tunnels) and with frontier homeomorphic to ...

**2**

votes

**1**answer

76 views

### Priestley topologizability and connected components

This question is in the spirit of another older question.
We say that a poset $(P,\leq)$ is Priestley-topologizable if there is a topology $\tau$ on $P$ such that $(P,\leq,\tau)$ is a Priestley ...

**-4**

votes

**0**answers

61 views

### Sequences and reflexivity [on hold]

Assume $X$ to be a real reflexive Banach space.
Why are sequential topological notions
topological notions ?
For ex :
sequentially closed set is closed,
sequentially compact set is compact,
and so ...

**-6**

votes

**0**answers

45 views

### Every compact space is Lindelof Spaces [on hold]

Kindly give me the converse proof of the Theorem " Every compact space is Lindelof Spaces" with an example.

**12**

votes

**1**answer

534 views

### What if homotopy were expanded to allow any connected space instead of [0,1]?

What would happen to homotopy theory if we used a more general definition of homotopy, based on general connected spaces rather than [0,1]?
Given continuous $f,g:X\to Y$, define $f$ and $g$ to be ...

**4**

votes

**1**answer

167 views

### The subbase theorem for total boundedness

The uniform space analogue of Alexander's subbase lemma on compact subbase is (As we know, Alexander subbase lemma can be used to prove Tychonoff's theorem) :
Let $(X,\mathcal{U})$ be a uniform ...

**2**

votes

**0**answers

61 views

### convergence of integral for each bounded function in probability

Let $\mu, \mu_1, \mu_2, \dots$ be random measures on
a Polish space (separable completely metrizable topological space) $(S, {\mathcal S})$.
Suppose I know that
$$\int f d \mu_n \to \int f d\mu$$
...

**2**

votes

**0**answers

67 views

### Non-compact and maximal non-$T_2$ [migrated]

Is there a space $(X,\tau)$ that is not compact, not $T_2$, but for every topology $\tau'\supseteq \tau$ with $\tau'\neq\tau$ the space $(X,\tau')$ is $T_2$?

**5**

votes

**2**answers

154 views

### Extending hyperconnected spaces

A hyperconnected space is a topological space such that every two non-empty open sets have non-empty intersection. Let's call a space $(X,\cal{T})$ maximally hyperconnected if it is hyperconnected and ...

**21**

votes

**2**answers

1k views

### Why should have Peter May worked with CGWH instead of CGH in “The Geometry of Iterated Loop Space”?

This is a follow-up to Dan Ramras' answer of this question.
The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here).
The weak Hausdorff rather ...

**3**

votes

**2**answers

102 views

### Minimal Hausdorffness reversed

It turns out that not every Hausdorff topology is contained in a minimal Hausdorff topology. Let's put this question on its head: is every non-$T_2$ topology contained in a topology that is maximal ...

**-1**

votes

**0**answers

51 views

### Generalization of a class of sets [closed]

In topological space, we start with open set, which serves as fundamental set. We know that union of finite disjoint open sets is the smallest set amongst any kind of unions of open sets, so we have a ...

**1**

vote

**0**answers

68 views

### Two questions on hyperspace of a metric space

Let $(X,T)$ be a compact metrizable space. For every metric $d$ on $X$ which is $T-$ compatible, the Hausdorff metric on $2^{X}$ gives a topology on $2^{X}$.
(Up to homeomorphism) is this topology ...

**2**

votes

**0**answers

47 views

### Is the order convergence topology on a poset always Hausdorff?

In this post two topologies on a poset $(P,\leq)$ were defined: the interval topology $\tau_i(P)$ and the order convergence topology $\tau_o(P)$. It turns out that $\tau_i(P)$ is always $T_1$ and ...

**6**

votes

**2**answers

144 views

### Does every locally compact Hausdorff space admit a locally finite open covering by relatively compact sets?

Let $X$ be a locally compact Hausdorff space. Does there exist a locally finite open covering consisting of relatively compact sets?

**2**

votes

**1**answer

75 views

### Interval topology and order convergence topology

Throughout this post, let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where ...

**2**

votes

**0**answers

79 views

### Mean on compact metric spaces

Let $X$ be a compact metric space. A $k$ mean on $X$ is a continuous map $f:X^{k}\to X$ which is identity on the diagonal and is invariant under all $k$-permutations. For details, See the following ...

**-1**

votes

**0**answers

42 views

### Between metacompactness and infinite Lebesgue covering dimension [migrated]

A space $(X,\tau)$ is said to be metacompact if every open covering $\cal U$ has a refinement $\cal V$ such that for every $x\in X$ the set ${\cal V}_x := \{V\in \mathcal{V}: x\in V\}$ is finite.
A ...

**2**

votes

**2**answers

176 views

### Two notions of zero-dimensionality for topological spaces

Let $(X,\tau)$ be a topological space.
We say that $(X,\tau)$ is zero-dimensional with respect to the Lebesgue covering dimension (zd1) if every open cover of the space has a refinement which is a ...

**1**

vote

**0**answers

35 views

### Reference request: Topological space of polygonal chains and its properties [migrated]

I'm interested in approximations of $C^1([a,b])$-functions by polygonal chains:
image File:NURBstatic.svg by User:WulfTheSaxon licensed under GNU Free Documentation License
A polygonal chain can be ...

**-1**

votes

**4**answers

365 views

### Studying topology: which first, algebraic or differential? [closed]

I have recently studying the basics of topology (ideas in point set, connectedness compactness) and I want to continue my studies but i'm interested in both differential and algebraic topology. which ...

**3**

votes

**1**answer

138 views

### Are all minimal totally separated spaces compact?

Let us call a space $(X,\tau)$ totally separated if for every two distinct points there is a clopen set containing one, but not the other. If for every topology $\sigma\subseteq\tau$ with $\sigma\neq ...

**2**

votes

**1**answer

127 views

### Minimal totally separated spaces

Let us call a space $(X,\tau)$ totally separated (ts) if for every two distinct points there is a clopen set containing one, but not the other. If for every topology $\sigma\subseteq\tau$ with ...

**1**

vote

**2**answers

82 views

### Connected, maximal compact, but not $T_2$

Is there a connected topological space that is maximal compact, but not $T_2$? (A space $(X,\tau)$ is said to be maximal compact if for any topology $\tau'$ on $X$ with $\tau'\supseteq \tau$ and ...

**1**

vote

**0**answers

70 views

### Equicontinuity of $\{f_{2n}\circ f_{2n-1}\}$

Let $(X,D)$ be a compact metric space and $\{f_n\}_{n\in\mathbb{N}}$ be a sequence of homeomorphisms of $(X,d)$. It is easy to see that if $\{f_n\}$ is uniformly convergent then $\{g_n\}$ defined by ...

**2**

votes

**1**answer

150 views

### Does hyperconnected imply path-connected

If a space is hyperconnected (that is, the every non-empty open sets intersect), is it also path-connected?

**3**

votes

**0**answers

82 views

### Does “equalisers always closed” imply $T_2$? [migrated]

Is there a non-$T_2$ space $(X,\tau)$ with the following property?
For all topological spaces $A$ and continous maps $f,g:A\to X$ the set $\{a\in A: f(a) = g(a)\} \subseteq A$ is closed.

**20**

votes

**3**answers

1k views

### Does the Brouwer fixed point theorem admit a constructive proof?

Wikipedia and a few websites (and a few mathoverflow answers) say there is a constructive proof of the Brouwer fixed point theorem, some others say no. The argument for a constructive proof is always ...

**4**

votes

**0**answers

63 views

### Topological systems of imprimitivity

Let $G$ be a group acting by homeomorphisms on a topological space $X$. $G$ is topologically transitive if every open $G$-invariant subset of $X$ is empty or dense.
Here is an attempt to define ...

**0**

votes

**1**answer

43 views

### Countable, $T_1$, and not metacompact

Is there a countable space that is $T_1$ and not metacompact? (A space $(X,\tau)$ is not metacompact iff there is on open cover $\cal{U}_0$ such that for every open refinement $\cal V$ there is $x\in ...

**-1**

votes

**1**answer

131 views

### Decomposition space of $\mathbb{C}$ by concentric circles [closed]

What are the topological properties of the quotient space $X$ obtained from $\mathbb{C}$ by identifying points of the same modulus? I.e., the space $X=\mathbb{C}/E$ where $E$ is the equivalence ...

**4**

votes

**1**answer

113 views

### Segments on a family of parallel lines

Let $\{l_i:i\in I\}$ be a family of parallel lines on the plane $\mathbb{R}^2$. Suppose for each $i\in I$ there is a closed segment $s_i\subset l_i$. Moreover, for each triple $i_1,i_2,i_3$ there ...

**6**

votes

**1**answer

216 views

### is this map a closed inclusion?

I apologize in advance if this question is too technical. I haven't found a reference in the literature yet, and it seems difficult enough that perhaps it has not been answered.
Let $A$, $B$, and $C$ ...

**0**

votes

**0**answers

121 views

### Snake-like continua and universal images

Among the Hausdorff compact spaces the closed interval is the simplest snake-like continuum. I'll present the definition after stating the problem.
The snake-like continua $\ S\ $ are universal ...

**4**

votes

**0**answers

104 views

### Are there any known ``topological" invariants for branched coverings?

My question is the following: let $f:\Omega\to \mathbb{R}^n$ be a branched covering, namely $f$ is continuous, discrete (each fiber is a discrete subset of $\Omega$) and open (open sets get mapped ...

**2**

votes

**2**answers

168 views

### A Jordan Separation Theorem for Polyhedral Surfaces

Let me begin by defining what a polyhedral surface is.
A path-connected subset $ P $ of $ \mathbb{R}^{3} $ is called a polyhedral surface iff it is the union of a finite collection $ \mathcal{C} $ of ...

**4**

votes

**1**answer

77 views

### Convergent filters generated by (not necessarily countable) chains

Suppose $\langle X,\mathscr{O}\rangle$ is a topological space and let $\mathscr{O}_x$ be the family of all open neighbourhoods of $x\in X$. Let $\mathscr{F}$ be the filter generated from ...

**2**

votes

**2**answers

167 views

### Connected complement manifold

I'm working on some problem in algebraic geometry. I need a reference to the following result:
Let $h\in\mathbb{N}$ with $h\geq1$ and let $F\in\mathbb{C}\left[x_{1},\ldots,x_{h}\right]$
be a non ...

**3**

votes

**1**answer

68 views

### Minimal $T_0$-spaces

As suggested by Joseph van Name in Is the associated order of a minimal $T_0$ space always total?, here's a natural question on $T_0$-spaces:
If $(X,\tau)$ is $T_0$, is there a minimal $T_0$ topology ...

**3**

votes

**4**answers

229 views

### Is there a compact connected Hausdorff space in which every non-empty $G_\delta$ set has non-empty interior?

Q1.
Is there a compact connected Hausdorff space (with at least two points) in which every non-empty $G_\delta$ set has non-empty interior? (Without the requirement for connectedness, every finite ...

**0**

votes

**1**answer

251 views

### Shrinking $\mathcal{C}_c^{\infty}(M)$ to obtain a first countable space

This is a follow-up to this question.
Let $M$ be manifold (Hausdorff, countable base, finite dimensional, if it simplifies anything embedded in $\mathbb{R}^n$).
I'm interested in the topological ...

**1**

vote

**2**answers

134 views

### Topological properties for which bijectively related imply homeomorphism

In this post I give examples of topological spaces for which bijectively relations imply existence of an homeomorphism. Namely:
Intervals of the real line.
Compact spaces.
I also give a ...

**1**

vote

**1**answer

121 views

### Does the countable $\sigma$-product of a separable Hilbert space have a first countable topology?

Let $\mathcal{l}^2$ be "the" separable real infinite dimensional hilbert space, e.g. the space of square-summable sequences of real numbers.
Let $\Box^{\mathbb{N}}\mathcal{l}^2$ be the countable ...

**2**

votes

**0**answers

62 views

### On compactness in $C(X)$

Let $X$ be a Tychonoff space. It is well known, that for a family of scalar functions equicontinuity + pointwise boundedness imply relative compactness in $C(X)$ (with compact-open topology). It is ...

**4**

votes

**1**answer

245 views

### Direct limit of compact topological spaces

Let $\{X_n\}_{n\in \mathbb{N}}$ be a direct system of compact topological spaces, meaning that we have morphisms $f_i\colon X_i \to X_{i+1}$ with the necessary compatibility conditions. Is there any ...

**9**

votes

**4**answers

396 views

### Between Tietze's and Dugundji's Extension Theorems

The celebrated Tietze Extension Theorem asserts that any continuous real-valued function defined on a closed subset of a normal space, can be extended to a continuous function on the whole space. Seen ...