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

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-2
votes
4answers
300 views

Studying topology: which first, algebraic or differential? [on hold]

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 ...
1
vote
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 ...
2
votes
2answers
100 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 ...
2
votes
1answer
110 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 ...
3
votes
0answers
111 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 ...
54
votes
5answers
3k views

How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?

This question is about the space of all topologies on a fixed set X. We may order the topologies by refinement, so that τ ≤ σ just in case every τ open set is open in σ. ...
31
votes
2answers
1k views

Continuous bijections vs. Homeomorphisms

This is motivated by an old question of Henno Brandsma. Two topological spaces $X$ and $Y$ are said to be bijectively related, if there exist continuous bijections $f:X \to Y$ and $g:Y \to X$. ...
36
votes
8answers
4k views

Non-homeomorphic spaces that have continuous bijections between them

What are nice examples of topological spaces $X$ and $Y$ such that $X$ and $Y$ are not homeomorphic but there do exist continuous bijections $f: X \mapsto Y$ and $g: Y \mapsto X$?
1
vote
2answers
76 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 ...
27
votes
1answer
1k views

Computing Self-Intersections with Complex Analysis

It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis: $$n = \oint_C\frac{dz}{z}.$$ You can also count the number of roots of $f(z) = 0$ inside a close ...
2
votes
1answer
140 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?
1
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0answers
64 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
1answer
125 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 ...
2
votes
1answer
75 views

Does “equalisers always closed” imply $T_2$?

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.
6
votes
1answer
207 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 $C$ ...
20
votes
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 ...
0
votes
0answers
99 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
0answers
61 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
1answer
41 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 ...
4
votes
1answer
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 ...
8
votes
1answer
527 views

Order homomorphism functions on $\omega_1$

Let $\omega_1$ be the first uncountable ordinal, same as the set of all countable ordinals. Let $F$ be the set of all functions $f$ from $\omega_1$ minus singleton $0$ into $\omega_1$ that are (a) ...
33
votes
9answers
4k views

understanding Steenrod squares

There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called Steenrod squaring: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from ...
5
votes
1answer
104 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 ...
4
votes
0answers
101 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 ...
1
vote
2answers
129 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 ...
4
votes
1answer
75 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 ...
0
votes
1answer
220 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 ...
2
votes
2answers
161 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
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 ...
3
votes
4answers
211 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 ...
1
vote
1answer
119 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
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
votes
1answer
235 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 ...
0
votes
1answer
39 views

Number of continuous characters on an infinite Hausdorff precompact abelian group with exponent $p$

Let $(G,\mathcal T)$ be an infinite Hausdorff precompact abelian topological group and let $G$ have exponent $p$ where $p$ is a prime number. Can it be proved that there are at least $p+1$ continuous ...
8
votes
3answers
332 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
1answer
137 views

Weak topology on a pre-Hilbert Space

Since there was essentially no answers on my previous question, I will ask a partial case of it, which is very easy to state. Let $\left(X,\left<\cdot,\cdot\right>\right)$ be an inner product ...
1
vote
1answer
273 views

Weaker form of irreducible surjections

An irreducible surjection is usually defined as a continuous closed surjective map $f:X\rightarrow Y$ such that if for some closed set $C\subset X$ one has $f(C)=Y$ then $C=X$. In my dissertation I ...
1
vote
3answers
155 views

Is the poset of all precompact group topologies on an abelian group $G$, order-isomorphic to $\operatorname{Sub}(\hat{G})$?

In this page, in abstract, it is claimed that the poset of all Hausdorff precompact group topologies on an abelian group $G$, is order-isomorphic to the the subgroup lattice of $\hat{G}$, the ...
2
votes
0answers
118 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 ...
2
votes
1answer
287 views

Lindelöf subsets of $P$-spaces

A completely regular topological space $(X,\tau)$ is called a $P$-space, if every $G_\delta$-subset of $X$ is open. (i.e $\tau$ is closed under countable intersection). Here we recall some special ...
1
vote
0answers
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 $X$? ...
6
votes
2answers
323 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
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 ...
2
votes
1answer
107 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 ...
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 ...
3
votes
1answer
118 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
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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votes
3answers
2k views

Finite Hausdorff spaces [closed]

Is a finite Hausdorff space necessarily discrete?
12
votes
4answers
386 views

Obtain any 3-manifold from repeating surgeries on knots in $S^3$

In Witten's “QFT and Jones Polynomials” paper, page 383, it states that: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) ...
1
vote
1answer
110 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 ...