**1**

vote

**0**answers

57 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 ...

**1**

vote

**0**answers

38 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

132 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?

**-4**

votes

**0**answers

63 views

### Graph theory and topology [on hold]

I have related topological ideals with vertex magic totallabeling in graph theory. Is there any possibility to relate vertex magic totallabeling with generalized topology in a very interesting way? ...

**2**

votes

**1**answer

56 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

64 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

172 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 ...

**-2**

votes

**4**answers

343 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

136 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

125 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

78 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

68 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

148 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

62 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

42 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

105 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

212 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

101 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

102 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

103 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

164 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

223 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

228 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

131 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

120 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

238 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

391 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 ...

**1**

vote

**0**answers

29 views

### Homotopy injection between the unit ball in the Euclidean n space and an n-dimensional metric AR

Let $D^n$ be the closed unit ball in $\mathbb{R}^n$. Given a compact, $n$-dimensional, AR(Absolute Retract) metric space $X$, must it happen that either $X$ embeds in $D^n$ or $D^n$ embeds in $X$?
...

**2**

votes

**1**answer

108 views

### Is every set of small measure contained in an open set of small measure with null boundary?

Let $\lambda( \cdot )$ denote Lebesgue measure on $[0,1]$. Let $(A_n)_{n=1}^\infty$ be a decreasing sequence of Borel subsets of $[0,1]$ such that $\bigcap_{n=1}^\infty A_n = \emptyset$. Given ...

**2**

votes

**2**answers

133 views

### Is the associated order of a minimal $T_0$ space always total?

Let's call a space $(X,\tau)$ minimal $T_0$ if it is $T_0$ and for every topology $\sigma\subseteq \tau$ with $\sigma \neq \tau$ we have that $(X,\sigma)$ is not $T_0$ any more.
We say $x\leq y$ in a ...

**3**

votes

**1**answer

119 views

### Is there a maximal connected Hausdorff space?

We call a space $(X,\tau)$ maximal connected, if it is connected, and for any topology $\sigma \supseteq \tau$ with $\sigma\neq \tau$, the space $(X,\sigma)$ is not connected.
Is there a maximal ...

**1**

vote

**1**answer

86 views

### Maximal connected topologies

We call a space $(X,\tau)$ maximal connected, if it is connected, and for any topology $\sigma \supseteq \tau$ with $\sigma\neq \tau$, the space $(X,\sigma)$ is not connected.
If $(X,\tau)$ is ...

**6**

votes

**2**answers

327 views

### Relating different topologies on $C^{\infty}_c(M)$

This is somehow connected to this question.
I can think of at least four topologies to put on $C_c(M)$:
Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney ...

**2**

votes

**0**answers

122 views

### Weak topology on subsets of a normed space

I have few questions about the subsets of a normed space $X$ endowed with the weak topology. Let $E$ be such subset.
When is the norm a continuous function on $E$?
When is the metric induced by the ...

**3**

votes

**0**answers

60 views

### Which compact topological spaces are homeomorphic to their ultrapower?

It is well known that for any compact metric space $(X, d)$, and any ultrafilter $\mu$ there is a map $i_\mu:\prod_\mu (X, d) \to (X_d)$ in the category of metric spaces and Lipschitz maps where ...

**7**

votes

**2**answers

516 views

### Why not develop a Hamiltonian-based Morse theory?

I have begun to learn the basics of Morse theory and Floer homology. I understand that Floer homology is the natural theory for symplectic manifolds, but from my preliminary knowledge of Morse theory ...

**1**

vote

**0**answers

38 views

### On compactness in Sion's minimax theorem

Sions minimax theorem (wiki, paper) can be stated as follows:
Let $X$ be a compact convex subset of a linear topological space and $Y$ a convex
subset of a linear topological space. Let $f$ be a ...

**1**

vote

**1**answer

112 views

### Bounded-open topology vs norm on $L\left(X,Y\right)$

In general topology there is two ways of introducing a topology on the space of (continuous) maps between, say, metric spaces: set-open topology and uniform topology (it is a uniformity of uniform ...

**1**

vote

**0**answers

54 views

### Is $\mathcal{P}(\omega)/fin$ with the interval topology path-connected?

Is $\mathcal{P}(\omega)/fin$ with the interval topology path-connected? (You find the definition of $\mathcal{P}(\omega)/fin$ here.)

**1**

vote

**2**answers

98 views

### Continuous image relation on topological spaces

Let $\kappa$ be a cardinal, and let $\text{Top}(\kappa)$ be the set of topological spaces $(X,\tau)$ such that $X\subseteq \kappa$. We pre-order $\text{Top}(\kappa)$ by
for $X, Y \in ...

**23**

votes

**4**answers

831 views

### Connectedness in the language of path-connectedness

Is there a topological space $(C,\tau_C)$ and two points $c_0\neq c_1\in C$ such that the following holds?
A space $(X,\tau)$ is connected if and only if for all $x,y\in X$ there is a ...

**5**

votes

**1**answer

106 views

### Is an open map with open relative diagonal necessarily a local homeomorphism?

Let $f : X \to Y$ be an open (and continuous) map of locales. Suppose the relative diagonal $\Delta_f : X \to X \times_Y X$ is an open embedding of locales. Does it follow that $f : X \to Y$ is a ...