Questions tagged [gn.general-topology]

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

Filter by
Sorted by
Tagged with
1 vote
0 answers
136 views

Complement of contractible locally Euclidean subspace

Let $X$ be a connected closed topological manifold. Let $S\subset X$ be a contractible locally Euclidean subspace. Is $X\setminus S$ connected?
1 vote
0 answers
152 views

Homotopy groups of ball complement

Let $X$ be a connected closed topological manifold. Let $n$ be an integer such that $\pi_i(X)=\{0\}$ for $1\leq i \leq n$. Let $f:B^m\to X$ be a topological embedding, where $B^m$ is the $m$-...
4 votes
0 answers
266 views

Sierpinski's characterization of $F_{\sigma\delta}$ spaces

According to [2]: Let $X$ be a space. We call a system $(X_s)_{s\in T}$ a Sierpinski stratification of $X$ if $T$ is a nonempty tree over a countable alphabet and $X_s$ is a closed subset of $X$ for ...
4 votes
1 answer
162 views

Consider a net of weak order units in a Riesz space converging in order to a weak order unit. Is there a tail whose infimum is a weak order unit?

Let $X$ be an extremally disconnected (the closure of an open set is open) compact Hausdorff space, and consider the Riesz space $C^\infty(X)$ of continuous functions from $X$ to the extended real ...
2 votes
1 answer
170 views

Set of null-sequences is not $\sigma$-compact

I am interested in a reference for the following fact (or a similar result). PROPOSITION. Let $X$ denote the set of real null sequences; i.e., the set of $(a_n)_{n=0}^{\infty}$ with $a_n\to 0$, with ...
6 votes
2 answers
986 views

Foundational results dependent on/equivalent to the continuum hypothesis or its negation?

I remember at a certain point early in my mathematical studies learning that the Axiom of Choice is equivalent to the following statement on Cartesian products: If $\{ X_i \}_{i \in I}$ is any ...
16 votes
2 answers
3k views

Is there a conceptual reason why topological spaces have quotient structures while metric spaces don't?

Of the mathematical objects that I am familiar with, it is normally the case that the product of 2 objects is an object of the same type and that an equivalence relation on an object induces a ...
18 votes
3 answers
4k views

Is a inverse limit of compact spaces again compact ?

Then one can construct a model for the inverse limit by taking all the compatible sequences. This is a subspace of a product of compact spaces. This product is compact by Tychonoff. If all the spaces ...
1 vote
1 answer
148 views

Regularity and ultrafilter

I read the following result in an article. Let $X$ be a regular space. Let $\mathcal{M}$ be free closed ultrafilter on $X$. Set $\mathcal{U=}\left\{ U:U\text{ is open and there exists a }F\in \mathcal{...
0 votes
1 answer
169 views

P-filter property?

Let $\mathcal{F}$ be a $P$-filter on $\omega$. Denote by $\Omega=\bigsqcup \omega_i$ where $\omega_i=\omega$. Consider the $P$-filter $\mathcal{S}$ on $\Omega$ whose base is as follows $(\bigsqcup_i ...
6 votes
2 answers
175 views

Nonhomeomophic spaces with homeomorphic mapping cones

It is natural to ask if it is possible for the mapping cone $X\cup_\alpha CA$ to be homeomorphic to the mapping cone $X\cup_\beta CB$ with $A$ and $B$ nonhomeomorphic. Is there a standard go-to ...
7 votes
1 answer
411 views

Proper homotopy

Let $F: X \times [0, 1] \to Y$ be a homotopy such that for any $t \in [0,1]$ the map $F( \cdot, t) : X \to Y$ is proper. Is it true in general that $F$ is proper? I am interested in particular in ...
1 vote
1 answer
270 views

Comparison of product topology and colimit topology in sequence spaces

In Munkres Theorem 20.4 it is shown that the (relative) uniform topology induced by: $$ d(x,y)\triangleq \sup_{n \in \mathbb{N}} d(x_n,y_n) $$ is strictly finer than the product topology on $\prod_{n \...
1 vote
1 answer
198 views

Is every homeomorphism approximately a product of homeomorphisms?

Let $\phi$ be a homeomorphism on $\mathbb{R}^{n+m}$, $\epsilon>0$, and $K\subseteq \mathbb{R}^n$ be a non-empty compact. Does there necessarily exist homeomorphisms $\phi_1,\phi_2$ on $\mathbb{R}^...
2 votes
1 answer
230 views

Approximate selection for finite-valued upper hemicontinuous/semicontinuous maps?

I'd like to know if there are any known-results on the existence of continuous approximation theorems for upper hemicontinuous (aka upper semicontinuous) maps $\phi: X\rightarrow Y$ which are finite ...
5 votes
2 answers
157 views

Local cross-sections for free actions of finite groups

Let $G$ be a finite group, let $X$ be a locally compact Hausdorff space, and let $G$ act freely on $X$. It is well-known that the canonical quotient map $\pi\colon X\to X/G$ onto the orbit space $X/G$ ...
11 votes
2 answers
497 views

Continuous version of the fundamental theorem of invariant theory for the orthogonal group

A standard result in the invariant theory of the orthogonal group states the following. Theorem Let $(E, \langle .,. \rangle)$ be an n-dimensional euclidean vector space, let $f : E^m \rightarrow {\bf ...
0 votes
0 answers
160 views

Problem of Thickening an Arc in a Topological $ 2 $-Manifold

Let $ M $ be a topological $ 2 $-manifold (possibly with boundary), $ C $ an arc in the interior of $ M $ (i.e., an injective continuous function from $ [- 1,1] $ into $ \operatorname{Int}(M) $), and $...
1 vote
0 answers
83 views

Compact metrizable contractible locally contractible topological space of finite covering dimension is a CW complex

Let $X$ be a compact metrizable contractible locally contractible topological space of finite covering dimension. Is $X$ homeomorphic to a CW complex?
5 votes
1 answer
445 views

Any continuous map is homotopic to one assuming fixed values at finitely many points

Let $X$ and $Y$ be topological spaces. Assume $X$ is locally contractible and has no dense finite subset. Assume $Y$ is path-connected. Given $n$ pairs of points $(x_i, y_i)$ where $x_i\in X$ and $y_i\...
1 vote
1 answer
70 views

Projecting Graph of a Function acted on by a homeomorphism

Let $X,Y$ be compact, connected, simply-connected, and separable, metric spaces each with at-least $2$-points, and let $f,g:X\rightarrow Y$ be continuous functions. Does there always exist a ...
2 votes
1 answer
103 views

When almost all points are not isolated in all subspaces

Let $X$ be a compact (non Hausdorff) $T_0$ topological space such that for any subset $\mathcal{A}=\{\mathfrak{x}_\alpha\}_{\alpha\in \Lambda}$ of distinct element of $X$ the set $\{\mathfrak{x}_\beta\...
7 votes
6 answers
2k views

How to partition R^3 into pairwise non-parallel lines?

Problem. How to partition R^3 into pairwise non-parallel lines? A possible solution is to stack infinitely many ``concentric'' hyperboloids, by increasing radius and decreasing slope. And don't forget ...
11 votes
1 answer
446 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. Question. If no point of X is a cut point, does X always have base of connected open sets whose complements (...
7 votes
0 answers
216 views

adding one point from the Stone-Cech compactification

Let $X$ be any non-compact Tychonoff space and $\beta X$ be its Stone-Čech compactification. The following fact is known: any point $p$ from the reminder $\beta X \setminus X$ is not a $G_{\delta}$-...
2 votes
0 answers
70 views

Separating a certain planar region with an open set

I have a fairly specific question related to plane separation properties. I couldn't quite see how to use Phragmen–Brouwer properties to answer it because those kind of results generally apply to ...
4 votes
0 answers
344 views

When every closed and connected subset is path connected

Let $X$ be a compact $T_0$ topological space such that its closed and connected subsets are path connected. Is there any characterization for such a space?
0 votes
2 answers
209 views

Intrinsically defining smooth/continuous/analytic functions

In mathematics, the notion of a continuous/smooth/analytic function $\mathbb{R}\to\mathbb{R}$ is introduced by defining the general set-theoretic function $\mathbb{R}\to\mathbb{R}$ and then imposing ...
2 votes
1 answer
186 views

Are there minimal topological conditions on a space 𝑋 for it to have a countable separating set?

Are there minimal topological conditions on a space $X$ for it to have a countable separating set? A separating set here is a set $D \subset C(X)$ (where $C(X)$ is the space of continuous functions ...
4 votes
0 answers
125 views

'Monodromy' for relative homology group

Let $A$ and $X$ be topological manifolds. Denote by $\mathbb {Emb}(A,X)$ the space of all topological embeddings $A\to X$. A loop $f_s:A\to X$ ($s\in[0,1]$) in $\mathbb{Emb}(A,X)$ should give rise to ...
2 votes
1 answer
68 views

Equicontinuity-like property of a convex compact set

Let $X$ be a Tychonoff topological space and let $x\in X$. Let $B\subset C(X)$ be convex and compact in the topology of pointwise convergence, and such that $f(x)=1$, for every $f\in B$. Is there an ...
5 votes
1 answer
338 views

Mapping $\mathbb P$ onto $\mathbb Q ^\omega$

Let $\mathbb P$ denote the space of irrationals. Is there a continuous bijection (one-to-one and onto) $f:\mathbb P\to \mathbb Q ^\omega$ that maps each closed subset of $\mathbb P$ to a $G_\delta$-...
4 votes
1 answer
296 views

Functoriality of Atiyah-Hirzebruch spectral sequence - Reference Request

I'm interested in a text book reference on the functoriality of the Atiyah–Hirzebruch spectral sequence. The only reference I found are these lecture notes by Kupers (link should lead to the target ...
0 votes
0 answers
45 views

Critical Growth of Dimension for Dense Cover by Linear Subspaces

Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that For any sequence of distinct finite-dimensional ...
5 votes
1 answer
204 views

Arc connectedness of product spaces

Is arc connected-ness well-behaved with respect to products? That is - $\prod X_\alpha$ is arc connected iff $X_\alpha$ is arc connected $\forall \alpha$ In this question on MathStackexchange, an ...
4 votes
1 answer
229 views

Measurable total order

Under what conditions on a metric space $X$, equipped with the Borel $\sigma$-algebra, does there exist a measurable total ordering of the elements of $X$? By "measurable total ordering" we ...
32 votes
1 answer
2k views

Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology?

On page 205 of his Topology textbook, James Munkres made an interesting remark: It is not known whether $\mathbb{R}^\omega$ is normal in the box topology. Mary-Ellen Rudin has shown that the answer ...
3 votes
2 answers
307 views

A minimal continuum

A continuum $X$ is called minimal if it is not a single point and is homeomorphic to all its nontrivial subcontinua. Here a trivial continuum is a single point. What is an example of a minimal ...
3 votes
0 answers
72 views

A holomorphic shrinking of a domain into a compact subset

This question is related to these two. Let $X\subset \mathbb{C}^{n}$ be a bounded domain. I am interested in the following property: there is a sequence of continuous maps $\varphi_n:\overline{X}\to X$...
-3 votes
1 answer
312 views

Basis for space of continuous, surjective monotone functions on $\mathbb{R}$ [closed]

$\DeclareMathOperator\CM{CM}$ I recently came across Okhezin - Study of families of monotone continuous functions on Tychonoff spaces describing monotone functions on general topological spaces and I ...
5 votes
0 answers
144 views

Do products of distance functions separate points?

Let $(X,d)$ be a metric space without isolated points and of diameter $1$. Let $Y=\{y_m\}_{m=1}^{\infty}$ be a dense subset of $X$. Define $g_0\equiv 1$, and for $m>0$ let $g_m=d(\cdot,y_1)\dotsm d(...
1 vote
0 answers
107 views

Is the Vietoris topology on compact subsets of $\mathbb R^n$ locally convex?

The title question says it all really. If the question is negative for compact subsets of $\mathbb R^n$, is it affirmative for compact and convex subsets of $\mathbb R^n$? How about for all nonempty ...
9 votes
0 answers
250 views

Does a generalization of Tietze's extension theorem hold for set-valued functions?

Let $X$ be a normal topological space. Tietze's extension theorem says that if $A \subset X$ is closed, then a continuous function $f: A \to \mathbb R^n$ can be extended to a continuous function whose ...
3 votes
0 answers
67 views

Is there a complex Katetov-Tong theorem?

For any bounded function $f: X \to \mathbb R$, not-necessarilly continuous, one can define for any $x$, the real functions $$ \limsup f(x) = \inf_{U\in\mathcal V_x} \sup_{u\in U} f(u) $$ and $$ \...
1 vote
1 answer
753 views

Known dense subset of Schwartz-like space and $C_c^{\infty}$?

After reading this question, which asked for some examples of commonly used (proper) dense subsets of $C_0^{\infty}(\mathbb{R}^n)$ with the $L^p$-norm I wonder. What are some "well-known" ...
0 votes
1 answer
75 views

Ultrabornological representation for the space of uniformly continuous functions?

Let $\{\omega_i\}_{i\in I}$ be a non-empty set of increasing (not necessarily strictly) continuous functions preserving $0$. Then, for each $i \in I$ define the space $$ C_{\omega_i}(\mathbb{R}^n,\...
3 votes
1 answer
139 views

Is each preseparable topological group narrow?

A topological group $G$ is defined to be $\bullet$ precompact if for any neighborhood $U\subseteq G$ of the unit there exists a finite subset $F\subseteq G$ such that $G=UF$; $\bullet$ narrow if for ...
2 votes
1 answer
102 views

Isomorphic hypergraphs of dense sets of Hausdorff spaces

If $H_i = (V_i, E_i)$ are hypergraphs for $i = 1,2$ , we say that they are isomorphic if there is a bijection $f:V_1 \to V_2$ such that for all $e\subseteq V_1$ we have $e\in E_1$ if and only if $f(e)\...
1 vote
2 answers
315 views

Connectedness of compact metric space

Let $X$ be a compact metric space satisfying the following condition: for any given positive number $\delta>0$, only finitely many components of $X$ have diameter larger than $\delta$. For a given ...
1 vote
0 answers
63 views

Connected components of bounded linear operators of $V = (\mathcal C(U(1), \mathbb C) , \lVert \cdot \rVert_\infty)$

This question is related to this one. Consider the complex Banach space $V=(\mathcal C(U(1), \mathbb C), \Vert \cdot \Vert_\infty)$ where $\mathcal C(U(1), \mathbb C)$ is the space of continuous ...

1
25 26
27
28 29
89