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

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14 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 ...
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0answers
43 views

Generalization of a class of sets [on hold]

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 ...
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0answers
64 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 ...
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0answers
39 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 ...
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2answers
140 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?
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63 views

Graph theory and topology [closed]

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
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1answer
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
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0answers
65 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 ...
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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
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2answers
173 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 ...
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0answers
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 ...
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4answers
351 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 ...
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1answer
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
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1answer
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
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2answers
79 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 ...
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0answers
69 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 ...
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1answer
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?
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0answers
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.
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3answers
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 ...
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0answers
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 ...
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1answer
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 ...
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1answer
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
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1answer
106 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
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1answer
214 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 $C$ ...
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0answers
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 ...
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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 ...
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2answers
115 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
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1answer
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 ...
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2answers
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
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1answer
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 ...
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4answers
224 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 ...
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1answer
235 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 ...
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2answers
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 ...
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1answer
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 ...
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0answers
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
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1answer
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 ...
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4answers
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 ...
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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
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1answer
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
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2answers
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
1answer
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
1answer
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 ...
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2answers
333 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 ...
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0answers
123 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 ...
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0answers
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 ...
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2answers
517 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 ...
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0answers
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 ...
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1answer
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 ...
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0answers
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.)
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2answers
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 ...