Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [gn.general-topology]

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

12
votes
1answer
374 views

When can I “draw” a topology in Baire space?

The motivation for this question is a bit convoluted, so in the interests of conciseness I'm just asking it as a curiosity (and I do find it interesting on its own); if anyone is interested, feel free ...
1
vote
1answer
43 views

A sequence in generalized order spaces

Let $X$ be a GO-space with the topology $\tau$ and $\lambda$ be the usual open interval topology on $X$. Put $$ R= \{x\in X: [x, \rightarrow) \in \tau\setminus \lambda \} \text{ and } L= \{x\in X: (\...
4
votes
1answer
100 views

Does every cut-point space embed into the plane?

Let $X$ be a connected separable metrizable topological space. Call it a cut-point space if $X\setminus \{x\}$ is disconnected for every $x\in X$. Then does $X$ embed into the plane? My thoughts: (...
4
votes
0answers
370 views

Questions about obstruction theory (Hatcher's book)

I'm actually studying obstruction theory as presented in the last section of chapter $4$ of the book Algebraic Topology by Allen Hatcher. He first finds condition so that a space $X$ admits a ...
1
vote
2answers
99 views

Extending homeomorphisms between compact metric subsets

Let $X$ be a compact metric, second countable space with finite covering dimension. Let $A,B$ be two closed subsets of $X$. Assume that $h:A\to B$ is a homeomorphism. Is it possible to extend $h$ to a ...
3
votes
1answer
121 views

Continuity concepts for correspondences

Consider two metric spaces (X,d) and (Y,d') and a correspondence F from X to Y. Does a topology on the power set of Y, P(Y) exists such that F is upper (resp. lower hemi- continuous) if and only if F ...
3
votes
1answer
157 views

Is there a metaLindelof nonLindelof space which has a dense hereditarily Lindelof subspace?

My question is as the title, i.e., Is there a metaLindelof nonLindelof space which has a dense hereditarily Lindelof subspace? The question is related to the following result: Every separable ...
2
votes
2answers
257 views

Paracompact zero-dimensional space without clopen partition refinement

If $(X,\tau)$ is a topological space we say that an open cover $\mathcal{U}$ is a clopen partition cover if it consists of disjoint clopen sets. Trivially, every clopen partition cover is locally ...
1
vote
0answers
93 views

How many two-dimensional space filling Hilbert-like curves are there?

I'm interested in filling 2d square with space filling, non-self-intersecting, locality preserving, self-similar curves, like Hilbert curve. I found interesting work concerning three dimensional case ...
11
votes
2answers
526 views

Two definitions of Lebesgue covering dimension

Maybe this question has already been considered here, but after a quick search I didn't find what I was looking for. As I see, in the literature there are two different definitions of the ...
22
votes
2answers
896 views

Which are the rigid suborders of the real line?

Which are the rigid suborders of the real line? If A is any set of reals, then it can be viewed as an order structure itself under the induced order (A,<). The question is, when is this structure ...
8
votes
3answers
671 views

How does the parity of $n$ affect the properties of $\mathbb{R}^n$? [closed]

Does the parity of the dimension of $\mathbb{R}^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?
6
votes
1answer
236 views

“Cyclic” continuum

On p. 221 of http://topology.auburn.edu/tp/reprints/v08/tp08113.pdf, I found the following definition: "A curve is said to be cyclic if its first Čech cohomology group with integer coefficients ...
2
votes
1answer
76 views

Is the following map a cofibration?

Assuming that the diagonal map $X\rightarrow X\times X$ is a cofibration. Is it true that the diagonal map $\Sigma X\rightarrow \Sigma X\times \Sigma X$ is a cofibration? (Where $\Sigma X$ is the ...
1
vote
0answers
99 views

Commutations of some limits and colimits in $\mathbf{CGWH}$

I know that finite limits do not commute with filtered colimits in general in $\mathbf{CGWH}$, nevertheless, do colimits commute with pullbacks, when we consider diagrams of the form $$\begin{matrix}&...
8
votes
1answer
287 views

Intersection of nested open ball in complete metric spaces is nonempty?

My question is that whether the following statement is true or not. In a complete metric space $(X, d)$, if a sequence of open balls $\{B(x_i, r_i)\}_{i=1}^\infty$ satisfies $$ \exists \epsilon > ...
0
votes
1answer
95 views

Continuity of maps in which preimage preserves compactness

Let $X$ and $Y$ be Hausdorff spaces and suppose that $Y$ is locally compact. Let $f:X\to Y$ be a surjective map such that for any compact subset $K \subset Y$ the pre-image $$f^{-1}(K)=\{x\in X: f(x)\...
1
vote
1answer
224 views

Sufficient conditions for a topological space to be regular $T_3$

There was a similar thread on the neighbour forum StackExchange on sufficient conditions for a topological space to be completely regular $T_{3^1/_2}$. Please, let me know any known condition(s) that ...
4
votes
1answer
184 views

Bounded growth of functions vs bounded growth of functions on countable sets

I am wondering if the boundedness of growth can be characterized by sequences. I am not sure if I use the term "growth" correctly, or use the correct tags for this question. Here is what I mean. Let $...
3
votes
1answer
84 views

Reduced suspension of a Hurewicz cofibration

I would like to know whether the reduced suspension of a Hurewicz cofibration of pointed spaces (it is a Hurewicz cofibration when considered as a map of unbased spaces) is an acyclic Hurewicz ...
4
votes
1answer
160 views

Embedding ordinals with the order topology into connected $T_2$-spaces

Is there a limit ordinal $\kappa_0$ with $\kappa_0 \lt 2^{\aleph_0}$ and such that for every limit ordinal $\lambda$ with $\kappa_0\leq \lambda\lt 2^{\aleph_0}$ there is a connected $T_2$-space $X_\...
5
votes
0answers
81 views

An easier example of complete lattice such that the Scott topology on it is not sober

Basic notions: $1$, A partially ordered set is a dcpo if each of its directed subsets has a supremum. (https://en.wikipedia.org/wiki/Complete_partial_order)\ $2$, A subset O of a dcpo P is called ...
2
votes
1answer
66 views

A monoidal model structure on pointed spaces

Do the classes of pointed Hurewicz cofibrations, pointed Hurewicz fibrations and pointed homotopy equivalences give a model structure on pointed (compactly generated weak Hausdorff) topological spaces ...
2
votes
1answer
85 views

Are separability and ccc equivalent for closed subspaces of $\beta N$?

Let $\beta \mathbb N$ be the Stone-Cech compactification of the integers. Then $\beta \mathbb N\setminus \mathbb N$ is non-separable because if fails the ccc condition, that is, it has an uncountable ...
5
votes
2answers
176 views

Codimension-1 subgroups of 3-manifold groups

Let $G$ be a finitely generated group and let $H$ be a subgroup of $G$. $H$ is a codimension-1 subgroup of $G$ if $C_{G}/H$ has more than one end, where $C_{G}$ is the Cayley graph of $G$. Do all ...
104
votes
3answers
7k views

Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
2
votes
1answer
79 views

Space which is $T_1$ and sober but not Hausdorff?

Every Hausdorff space is $T_1$ and sober. Does the converse hold? I expect not. What's a counterexample? I expected I should be able to look this up in Counterexamples in Topology, but unfortunately ...
1
vote
2answers
80 views

Can the Boolean Algebra of regular open sets be isomorphic to ${\cal P}(\omega)/(\text{fin})$?

Let $(X,\tau)$ be a topological space. $A\subseteq X$ is said to be regular open if $A = \text{int}(\text{cl}(A))$ and let $\text{RO}(X,\tau)$ denote the collection of regular open sets of $X$. A ...
4
votes
2answers
112 views

Inverse image of rational values

I am a postgraduate student of physics. While doing some research on Poincare's work on the integrability of the three body problem, I came up with the following problem (which I feel unable to handle,...
2
votes
1answer
187 views

Embedding into $C\times [0,1]$

Every totally disconnected separable metric space of dimension $n$ homeomorphically embeds into $C\times \mathbb R ^n$. Is something like this known? $X$ is totally disconnected means that every ...
1
vote
1answer
118 views

The completeness of spaces of continuous functions with the compact-open topology

For a Tychonoff space $X$ let $C_k(X)$ denote the space of continuous real-valued functions on $X$, endowed with the compact-open topology. Problem. Is the space $C_k(X)$ Polish if it is Polishable ...
16
votes
1answer
511 views

Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ with injective restriction $f|\mathbb Q^\omega$?

Question. Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ whose restriction $f|\mathbb Q^\omega$ is injective?
25
votes
3answers
730 views

Does $M^o=N^o$ imply that $\partial M = \partial N$?

let $M$ be a smooth $n$-manifold with boundary $\partial M$; I denote by $M^o$ the internal part of $M$, that is $M \smallsetminus \partial M$. The question is the same as in the title: let $M$ and $N$...
3
votes
1answer
65 views

Is the compact-open topology on the dual of a separable Frechet space sequential?

Let $X$ be a separable Frechet space (= Polish locally convex linear metric space) and $X'_c$ be the space of linear continuous functionals on $X$, endowed with the compact-open topology (= the ...
1
vote
1answer
61 views

Why is a certain space of linear isometries paracompact

Let $V$ be a finite dimensional real inner product space and $U$ a real inner product space of countable dimension. Why is the space of linear isometries from $V$ to $U$ paracompact?
2
votes
1answer
105 views

Commutation of filtered colimits and finite limits in $\mathbb{CGWH}$

Do filtered colimits and finite limits (in particular pullbacks) commute in the category of compactly generated weak Hausdorff spaces?
11
votes
0answers
160 views

Is homeomorphism of simplicial complexes semidecidable?

Conventions: $\cong$ is homeomorphism of topological spaces and isomorphism of groups, $\equiv_G$ is the equality of two words over the generators of the group $G$. Simplicial complexes are finite. ...
-2
votes
2answers
152 views

A countable polish space must be discrete? [closed]

I am looking for an elegant proof of the fact that a countable metric space is complete iff its underlying topology is discrete. It is easy to see that a discrete space is complete because its ...
7
votes
1answer
155 views

$2$-determined Hausdorff spaces

Is there an infinite Hausdorff space $(X,\tau)$ with the following property? If $x\neq y \in X$ and $f:\{x,y\}\to X$ is a map, then there is exactly one continuous function $f': X\to X$ such that $...
1
vote
1answer
133 views

Is the boundary of an open set in a $\sigma$-space empty?

Recall that a Boolean space is a $\sigma$-space in case the closure of every open Borel set is open. Let $\{B_i\}$ be a denumerable family of open-closed sets in a $\sigma$-space $X$. Then $\bigcup_i ...
9
votes
1answer
283 views

Is every metric continuum almost path-connected?

The question was motivated by this question of Anton Petrunin. By a metric continuum we understand a connected compact metric space. Let $p$ be a positive real number. A metric continuum $X$ is ...
3
votes
1answer
65 views

Intersecting geodesics on a surface from non-intersecting geodesics

Let $a$ and $b$ be non-intersecting closed geodesics on a hyperbolic surface. Can these curves be homotopied to transversely intersect but still be geodesics?
17
votes
3answers
1k views

“Anti” fixed point property

Let $(X,\tau)$ be a topological space. If $f:X\to X$ is continuous, we say $x\in X$ is a fixed point if $f(x) = x$. The space $(X,\tau)$ is said to have the anti fixed point property (AFPP) if the ...
1
vote
2answers
84 views

Large discrete subsets of connected $T_2$-spaces

If $(X,\tau)$ is a topological space, we say $S\subseteq X$ is discrete, if the subspace topology on $S$ inherited from $(X,\tau)$ is discrete. Let $\kappa$ be an infinite cardinal. Is there a ...
12
votes
1answer
353 views

Is the identity function a unique multiplicative homeomorphism of $\mathbb N$?

Endow the set $\mathbb N$ of positive integers with the topology $\tau$ generated by the base consisting of arithmetic progressions $a+b\mathbb N_0$ where $\mathbb N_0=\{0\}\cup\mathbb N$, where $a,b\...
9
votes
1answer
230 views

Set of homeomorphic fixed points that is dense, but not equal to whole space

If $(X,\tau)$ is a topological space, let $FH(X)$ denote the collection of $x\in X$ such that there is a non-identity homeomorphism $\varphi:X\to X$ with $\varphi(x) = x$. What is an example of a $...
6
votes
0answers
136 views

Quotients of 4-sphere by smooth $Z_p$ actions with knotted fixed point sets

This question is closely related to another I asked today. Giffen showed in 1966 that the generalized Smith conjecture is false by constructing for odd $p$ a smooth $Z_p$ action on $S^4$ with fixed-...
6
votes
0answers
95 views

Spatiality of products of locally compact locales

In Johnstone´s Sketches of an Elephant Volume 2, page 716, lemma 4.1.8 states that for spatial locales $X$ and $Y$ with $X$ locally compact then the locale product $X\times Y$ is spatial. Is this ...
11
votes
3answers
834 views

If Q is a subset of the plane of size less than continuum, then does every closed F in Q extend to a closed connected G in the plane with the same trace on Q? (Or is this independent of ZFC?)

This question arises in connection with this MO question and especially with Sergei Ivanov's wonderful answer, which showed that for any countable set $Q\subset\mathbb{R}^2$ and every closed set $F\...