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

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12
votes
1answer
189 views

Which sequential colimits commute with pullbacks in the category of topological spaces?

This question was asked on math.stackexchange.com without a reaction. Given diagrams of topological spaces $$X_0\rightarrow X_1\rightarrow\ldots$$ $$Y_0\rightarrow Y_1\rightarrow\ldots$$ ...
5
votes
1answer
124 views

Properties of the interval topology of the lattice of functions

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 $\downarrow x = \{y\in P: ...
1
vote
1answer
45 views

Does order-preserving equal continuous? [on hold]

Let $P,Q$ be posets and endow them with the interval topology $\tau_i(P)$ and $\tau_i(Q)$ respectively. Is it true that if $f: P\to Q$ is order-preserving, then it is continuous, and vice versa?
3
votes
0answers
29 views

Path-connected Hausdorff interval topologies

Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected and $T_2$. Does this imply that $[0,1]$ order-embeds into $P$? (This is a follow-up ...
9
votes
1answer
229 views

Meager subgroups of compact groups

Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre. Can $H$ always be covered by a countable family of nowhere dense sets $H_n$ such that $H_n^2$ is ...
5
votes
1answer
69 views

Path-connected interval topologies

Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected. Does this imply that $[0,1]$ order-embeds into $P$?
1
vote
0answers
170 views

Density of subspace with nonlocal/Wentzell boundary condition

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by ...
7
votes
3answers
549 views

Countable path-connected Hausdorff space

Is there a topology $\tau$ on $\omega$ such that $(\omega,\tau)$ is Hausdorff and path-connected?
2
votes
1answer
217 views

Homotopy with non piece-wise linear boundary

in the middle of a long proof I encounter the following problem. Let $E$ be a closed and convex set in $\mathbb R^n$ such that for all $\vec x\in E$ it holds that $\sum_ix_i=1$. (We can understand ...
0
votes
1answer
56 views

Corresponding between prime ideals in $C(X)$ and $C^*(X)$

we know that every maximal ideal in $C(X)$ is in this form: $$M^p=\left\{\,f \in C^*(x):\ p\in cl_{\beta X} Z\left(f\right)\,\right\}$$ and every maximal ideal in $C^*(X)$ is ...
3
votes
0answers
84 views

Cardinality based results in Topological Vector Spaces?

Given a topological vector space $V$, let its density be the smallest cardinal $A$ such that a set of cardinality $A$ is dense in $V$. Naively, it seems one of two things happen: TVS's $V$ of ...
13
votes
1answer
394 views

On the global structure of the Gromov-Hausdorff metric space

This is a purely idle question, which emerged during a conversation with a friend about what is (not) known about the space of compact metric spaces. I originally asked this question at ...
3
votes
1answer
97 views

Quotients of powers of the Sierpinski space

Is every space isomorphic to some quotient of a power of the Sierpinski space? More precisely: Let $(X,\tau)$ be a topological space, and let $\mathbb{S} = (\{0,1\}, \{\emptyset, \{0\},\{0,1\})$ be ...
-2
votes
0answers
38 views

interior, closure, continuity, composite in topology [closed]

Show by a counter example that the function f which assigns to each set is interior does not commute with the function g which assigns to each set is closure.
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. ...
2
votes
0answers
62 views

Quotients of simplicial complexes which are simplicial complexes

In the category of topological spaces, I would like to know that quotients of simplicial complexes (or $\Delta$-complexes) by equivalence relations which are "unramified" in a suitable sense still ...
4
votes
1answer
101 views

Approximation of topological dynamical systems?

I'm trying to find references to approximations of topological dynamical systems in the following sense: A topological dynamical system $(X, f)$ consists of a topological space (typically compact ...
1
vote
0answers
153 views

Stone-Cech compactification of $\mathbb{R}^n$ and smooth functions

I am currently attending a course where we are now covering the Stone-Cech compactification. Today we proved in some detail that extensions of bounded smooth functions on $\mathbb{R}^n$ to ...
3
votes
0answers
41 views

Injectively rigid spaces

Given a set $X$, is there a topology $\tau$ such that the identity $\text{id}_X$ on $X$ is the only continuous injective self-map? (This is Joel David Hamkins's recent question in the category ...
15
votes
3answers
429 views

Does every set $X$ have a topology for which the only continuous self-surjection is the identity map?

This question is a special case of Dominic van der Zypen's question Reconstructing relations with the image relation of a topology, as discussed in the comments, particularly the comment of Eric ...
6
votes
1answer
610 views

Maximal ideals of the rings of Baire-One Functions

A real function $f:X\rightarrow \mathbb{R}$ is called Baire-one function, if there is a sequence $(f_n)_{n=1} ^\infty$ of continuous functions $f_n:X\rightarrow \mathbb{R}$ on $X$ so that for all ...
5
votes
1answer
158 views

Implications between different covering properties of spaces

Let $X$ be a set. A set ${\cal C}\subseteq {\cal P}(X)$ is said to be a cover of $X$ if $\bigcup {\cal C} = X$ and $X\notin {\cal C}$. If ${\frak U}$ and $\frak{W}$ are collections of covers of a ...
3
votes
1answer
225 views

Strong measure zero sets and selection principles

A set of reals $X$ is strong measure zero if for any sequence of positive real numbers $ ( \epsilon_n ) _{n \in \omega }$ there is a sequence of open intervals $ ( a_n ) _{n \in \omega }$ which ...
10
votes
2answers
1k views

Does Riemann map depend continuously on the domain?

I've always taken this for granted until recently: In the simplest case, given Jordan curve $C \subseteq \mathbb{C}$ containing a neighborhood of $\bar{0}$ in its interior. Given parametrizations ...
10
votes
0answers
304 views

Linearly Lindelöf spaces with Lindelöf degree of uncountable cofinality

A space is linearly Lindelöf iff every open cover $C$ has a subcover $S$ with $\operatorname{cf} (|S|)= \aleph_{0}$. Is there a linearly Lindelöf space $X$ with $\operatorname{cf} (L(X))> ...
12
votes
0answers
511 views

Characterization of Fréchet-Urysohn spaces using sequential continuity at a point

A map $f \colon X \to Y$ is called sequentially continuous at the point $a$ if for every sequence $(x_n)$ such that $x_n\to a$, we also have $f(x_n)\to f(a)$. $$x_n\to a \qquad \Rightarrow \qquad ...
20
votes
0answers
1k views

Subfields of $\mathbb{C}$ isomorphic to $\mathbb{R}$ that have Baire property, without Choice

While sitting through my complex analysis class, beginning with a very low level introduction, the teacher mentioned the obvious subfield of $\mathbb{C}$ isomorphic to $\mathbb{R}$, and I then ...
23
votes
1answer
855 views

A group allowing exactly 7 group topologies

Is there a group $G$ allowing exactly 7 group topologies on $G$: $\mathcal T_{\text{trivial}}, \mathcal T_{\text{discrete}}, \mathcal T_1, \mathcal T_2,\mathcal T_3,\mathcal T_4, \mathcal T_5$ with ...
13
votes
1answer
441 views

Avoiding countable subgroups of a group homeomorphic to the Cantor space

The following question is motivated by the paper [Brian, Mislove, Every compact group can have a non-measurable subgroup]. A positive solution to a variation of the following problem implies a ...
16
votes
1answer
2k views

Example of fiber bundle that is not a fibration

It is well-known that a fiber bundle under some mild hypothesis is a fibration, but I don't know any examples of fiber bundles which aren't (Hurewicz) fibrations (they should be weird examples, I ...
8
votes
0answers
300 views

Standard model structures on $Top$

Call a model structure on $Top$ (the category of topological spaces) standard, if the weak equivalences are the weak homotopy equivalences. In this nLab page, two standard model structures on $Top$ ...
-1
votes
0answers
66 views

diagonal stratum in symmetric product

Fix a Riemann surface $(\Sigma, j )$ and a partition $\pi: r=\Sigma a_in_i$ for integer $a_i \ge 1, n_i \ge 1$, there is a diagonal stratum $\chi_{\pi}$ indexed by $\pi$ comprising the image of the ...
15
votes
1answer
314 views

The Gelfand duality for pro-$C^*$-algebras

The Gelfand duality says that $$X\to C(X)$$ is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...
0
votes
1answer
47 views

Rank of a generall linear group over a finite field [closed]

What is the rank (minimal number of group generators) of the group $GL(n,F)$, when $F$ is a finite field of odd order? I found that $SL(n,F)$ is $2$, but I can't find this information.
5
votes
0answers
123 views

Topological Subset Take-Away

David Gale's subset take-away game is a game where two players A and B play with a finite set $S$. Players alternately choose proper nonempty subsets of $S$ such that if a subset is chosen, then none ...
4
votes
1answer
117 views

Reconstructing relations with the image relation of a topology

For any topological space $(X,\tau)$ we define $$R_{im}(X,\tau) := \{(x,y)\in X^2: (\exists f:X\to X) \text{ continuous and surjective with } f(x) = y\}.$$ Clearly, $R_{im}(X,\tau)$ is reflexive. This ...
3
votes
1answer
52 views

Hausdorff spaces with asymmetric image relation

For any topological space $(X,\tau)$ we define $$R_{im}(X,\tau) := \{(x,y)\in X^2: (\exists f:X\to X) \text{ continuous and surjective with } f(x) = y\}.$$ Clearly, $R_{im}(X,\tau)$ is reflexive, and ...
6
votes
1answer
158 views

Is there a universal $\omega$-limit set?

For the purposes of this question, a dynamical system means a compact metric space $X$ together with a continuous map $f: X \to X$. For $x \in X$, the $\omega$-limit set of $x$, denoted $\omega(x)$, ...
2
votes
1answer
76 views

“Universal” connected spaces

Let $\kappa$ be an infinite cardinal. Does there exist a topology $\tau_{\kappa+1}$ on $\kappa+1$ such that for any topological space $(X,\tau)$ with $|X|=\kappa$ the following statement is true? ...
4
votes
4answers
248 views

When is the boundary of an open planar set a Jordan curve?

Is the boundary of an open, regular, bounded, path-connected, and simply connected set a Jordan curve? Trying to find weakest condition on an open bounded set to apply Carathéodory's theorem. My ...
2
votes
1answer
46 views

Shrinkable decompositions with uncountably many non-degenerate elements?

Let $\mathcal D$ be an upper semicontinuous decomposition of $\mathbb S^n$ and let $\mathcal D'\subset\mathcal D$ be the set of non-singletons. The decomposition space $^{\mathbb S^n}/_{\mathcal D}$ ...
8
votes
4answers
1k views

When is a finite cw-complex a compact topological manifold?

I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact ...
10
votes
1answer
300 views

Connected but no path connected components

Is there a Borel subset of plane which is connected but whose only path connected components are singletons? I know that a Bernstein set is a non Borel example of such a set. Thanks!
3
votes
0answers
39 views

Is the covering property $\Omega \choose \text{T}$ closed under products?

Suppose that $(X_i)_{i\in I}$ is a family satisfying the covering property $\Omega \choose \text{T}$ (for the definition of this covering property, see this post). Does $\prod_{i\in I} X_i$ ...
125
votes
3answers
6k views

Is $\mathbb R^3$ the square of some topological space?

The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to $X \times X$ for some space $X$. $\mathbb{R}$ is not such a space: If $X ...
1
vote
0answers
81 views

When Max(R) is Hausdorff space? [duplicate]

Let $R$ be reduce commutative ring with identity (a commutative ring such that $a^n$=0 ($a\in R$) implise $a=0$) and $Max(R)$ be the set of all maximal ideals of $R$. The hull-kernel (or Zariski ...
6
votes
0answers
60 views

Hausdorff spaces with lattice isomorphism between the topologies [closed]

For $i=1,2$ let $(X_i,\tau_i)$ be Hausdorff spaces such that the lattices $\tau_1, \tau_2$ are isomorphic. Does this imply that $(X_1, \tau_1) \cong (X_2,\tau_2)$? (This is a follow-up question to ...
2
votes
1answer
43 views

$T_2$-spaces such that the lattices of open sets can be embedded into each other

Let $(X,\tau), (Y,\sigma)$ be $T_2$-spaces such that there are injective lattice homomorphisms $f: \tau\to \sigma$ and $g:\sigma\to \tau$. Does this imply that $(X,\tau)\cong (Y,\sigma)$?
8
votes
0answers
134 views

Why would we a priori expect $V(I)$ to satisfy axioms to define the closed sets for a topology on $\text{Proj}(S)$?

Reposted from math.stackexchange here. The topological space $\text{Proj}(S)$ has the underlying set$$\text{Proj}(S) = \{\mathfrak{p} \text{ a homogeneous prime such that }S_+ \not\subseteq ...
18
votes
2answers
900 views

Non-homeomorphic spaces such that taking away a point makes them homeomorphic

Are there topological spaces $X,Y$, each having more than $2$ points, such that $X\not\cong Y$, and there is a bijection $\varphi: X\to Y$ such that for all $x\in X$ the spaces $X\setminus \{x\}$ ...