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

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3
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85 views

How Universal is the Topological $\mathbb K$-algebra $C(\Omega, \mathbb K)$?

For $\Omega$ an arbitrary set the family $C(\Omega, \mathbb K)$ of all functions $\Omega \to \mathbb K$ becomes a complete topological $\mathbb K$-algebra under the topology of uniform convergence. ...
5
votes
1answer
261 views

Is $\beta \mathbb{D}\setminus \mathbb{D}$ a group?

NB the original question asked about $\beta\mathbb{D}$ rather than the corona, hence some of the initial comments. Is there a group operation on $\beta \mathbb{D} \setminus \mathbb{D}$ extending ...
12
votes
0answers
232 views

A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?

Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...
1
vote
1answer
75 views

Open cover not containing a certain subcover

Is there an infinite topological space $(X,\tau)$ with the following property? There is an open cover ${\cal U}^*$ such that $X\notin {\cal U}^*$; every finite subset $F\subseteq X$ is contained in ...
3
votes
2answers
389 views

Cup product of cohomology in a Serre spectral sequence

How to use Serre spectral sequence to compute cup product structures? Let $F\to E\to B$ be a fibration. Suppose all the differentials of the corresponding Serre spectral sequence of cohomology are ...
8
votes
2answers
595 views

Surreal compactness

In a comment here, Joel David Hamkins said: ...I think perhaps every set-sized open cover of a bounded interval in the surreals has a finite subcover, but there are proper class open covers with ...
3
votes
1answer
167 views

Universal covering and double cover functors

Initially posted on MSE Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}_*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to ...
7
votes
1answer
98 views

Characterisations of closed embeddings in $Top_1$?

Let $Top_1$ be the category of topological spaces which are $T_1.$ I am curious as to whether there is a categorical definition of what a closed embedding is in this environment. With a ...
13
votes
3answers
470 views

How bad can a circle domain get?

Let $X$ be a domain in the Riemann sphere $\widehat{\mathbb{C}}$. We say that $X$ is a circle domain if every connected component of its boundary is either a circle or a point. It was conjectured by ...
6
votes
0answers
191 views

Remote points in $\beta X$

It is known that in general convergence by sequences is not enough to account for all points in $\beta X \setminus X$, where $\beta X$ refers to the Stone-Cech compactification of a topological space $...
4
votes
1answer
113 views

Symmetry of a distance metric for a generating set of Topology

I was trying to prove that $\epsilon$-balls defined based on the shortest travel-time distance in a transportation network is a valid generating set for a topology of points on a transportation ...
3
votes
2answers
602 views

Banach algebraic proof of the Borsuk Ulam theorem

I am wondering whether there exists a proof of the classical Borsuk Ulam theorem for the Euclidean n-sphere, $n>2$ that is based only on the theory of Banach algebras. I checked on MR but had no ...
4
votes
1answer
97 views

On continuous perturbations of functions of the first Baire class on the Cantor set

Is it true that for any function of the first Baire class $f:X\to\mathbb R$ on the Cantor cube $X=2^\omega$ there is a continuous function $g:X\to[0,1]$ such that the image $(f+g)(X)$ is disjoint with ...
15
votes
1answer
427 views

A problem of Keisler and Tarski

The following question dates back to Keisler and Tarski: From accessible to inaccessible cardinals, Fund. Math. 53, 1964 and also perhaps Mazur: On continuous mappings of Cartesian products, Fund. ...
5
votes
0answers
137 views

Uniform approximation of separately continuous functions on zero-dimensional spaces

For topological spaces $X,Y,Z$ а function $f:X\times Y\to Z$ is called separately continuous if for any $(x,y)\in X\times Y$ the restrictions of $f$ to the sets $\{x\}\times Y$ and $X\times \{y\}$ are ...
9
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0answers
186 views

Embedding $\beta\mathbb{N}$ into a product of Cantor sets

Let us consider $\beta\mathbb{N}$, the Stone-Czech compactification of the natural numbers (where we do not take $0$ to be a natural number, so the only idempotent elements are nonprincipal ...
5
votes
0answers
154 views

On a very weak notion of fibration (of topological spaces)

Suppose that $f:Y \to X$ is a map of topological spaces, and lets assume for simplicity that $X$ is connected. For the fibers of $f$ to compute the homotopy fibers, one would usually want to demand ...
2
votes
0answers
76 views

Given locally compact and $\sigma$-compact, can we get partition of unity?

Let $X$ be a locally compact, $\sigma$-compact Polish (complete and separable metric) space. How to prove: "There is an increasing sequence of continuous cut-off functions with compact support, $0\...
3
votes
1answer
148 views

Continuous and open image of a Polish space

Suppose that we have a continuous open and closed surjection $f\colon X\to Y$ of a Polish space $X$ to $Y.$ The closeness of $f$ implies that $Y$ is a metric space. But i do not know how to use ...
3
votes
2answers
240 views

Is there an easier proof to show that the closed convex hull of a normalized weakly null sequence is weakly compact?

In a paper that I am reading there is a following step: Let $X$ be a Banach space and let $(x_k) \subset X$ be a normalized sequence that converges weakly to $0$. Then $\overline{co}(x_k)$ is a ...
4
votes
0answers
597 views

Examples of a topological semidirect product

Let $G$ be a compact topological group, and $\operatorname{Aut}(G)$ the group of autohomeomorphisms of $G$. I have proved some (topological) results about the holomorph $G\leftthreetimes \operatorname{...
1
vote
2answers
101 views

Retract embedding of $S^{n}$ in its unit tangent bundle

Acording to the comment of Mark Grant and the answer of Ryan Budney, I revise the question: For what even $n$, there is a retract embedding of of $S^n$ in its unit tangent bundle?
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0answers
110 views

Examples of value quantales

In his paper "Quantales and continuity spaces" R. C. Flagg gives the following examples of value quantales: the lattice $\bf{2}$ of truth values with usual addition, the lattice $\mathbb{R}_{+}$ of ...
8
votes
1answer
224 views

Is every locally compactly generated space compactly generated?

[Parse it as (locally compact)ly generated.] I stumbled across this one whilst supervising an undergraduate thesis. Convenient categories for homotopy theory (e.g. CGWH) have been discussed here ...
3
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0answers
138 views

Which Topological Spaces are Powers?

Given a topological space $X$ and closed subspace $Y \subset X$, it may be the case that $X$ is a power of $Y$. That means $\displaystyle X = \prod_{i < \kappa} Y_i$ for some cardinal $\kappa$ ...
5
votes
5answers
662 views

Must uncountable compact Hausdorff spaces have large discrete subsets?

The situation is this. I have a space $X$ which is second countable, compact, and Hausdorff (it's a modified form of a type space, though I don't think that matters here). It has size continuum. It ...
1
vote
0answers
85 views

Surjectivity of maps between spheres [closed]

I am wondering how to prove that a non-zero degree map from $S^n \to S^n$ is surjective. For example, identifying $S^1 \subset \mathbb{C}$, we can take $f:S^1 \to S^1$ via $f(z) = z^k$ with $k\neq 0$. ...
4
votes
4answers
501 views

When is the quotient by an $n$-fold loop space an $m$-fold loop space?

Given a map of $n$-fold loop spaces $X\to Y$, we can take the homotopy cofiber, denote it $Y/X$ (all spaces here will also have a base point, and all maps pointed). I have some basic questions about ...
2
votes
2answers
111 views

Special filters in the algebra of regular open sets of a topological space

Let $\mathrm{r}\mathscr{O}$ be the family of open domains (regular open sets) of a topological space $\langle X,\mathscr{O}\rangle$, that is: $$A\in\mathrm{r}\mathscr{O}\iff A=\mathrm{int(\mathrm{cl(A)...
1
vote
1answer
77 views

Similarity graph for continuous maps between Hausdorff spaces

Let $X, Y$ be topological spaces and $f,g: X\to Y$ continuous. Then we say that $f, g$ are similar if for all $V\subseteq Y$ open we have either $f^{-1}(V) = g^{-1}(V) = \emptyset$, or $f^{-1}(V) \...
-1
votes
1answer
210 views

An infinite set in a compact space

Let $X$ be a topological space. Is there any characterization for the property that says "for every infinit subset $A$ of $X$ there exists $a\in A$ such that if $f$ be an arbitrary real continuous ...
0
votes
1answer
77 views

When is any convergence sequence is stationary?

Is there any characterization for a topological space under which every convergent sequence is stationary? (e.g. discrete topology)
3
votes
0answers
404 views

Some counter examples in group theory

In this question, which we flag it as a community wiki question, we search for a big list of groups $G$ which can not be isomorphic to a structure mentioned in $i.$ for some $i \in \{1,2,\ldots,...
2
votes
1answer
144 views

Intersection of compact sets in the compact-open topology

Let $(X,\tau)$ be a topological space. We topologize $\tau$ itself in the following way. For $K\subseteq X$ compact, we set $${\cal V}_K=\{U\in \tau: U \supseteq K\}.$$ The compact-open topology on $\...
6
votes
0answers
118 views

Cutting a piece of cake that $n$ people value as exactly $w$

Stromquist and Woodall (1985) study the problem of Sets on which several measures agree. There are $n$ non-atomic value measures on the unit circle, and a parameter $w\in(0,1)$. The goal is to find a ...
5
votes
0answers
116 views

Is a successor to a successor to the trivial group topology on an Abelian group, totally bounded?

Is there an example of an Abelian group $G$ and group topologies $\cal S$ and $\cal T$ on it such that $\cal S$ is an immediate successor to the trivial topology on $G$ (i.e there is no other group ...
5
votes
0answers
64 views

Does the $D$-property have universal objects?

A space $(X,\tau)$ is called a $D$-space if whenever one is given a neighborhood $N(x)$ of $x$ for each $x\in X$, then there is a closed discrete subset $D\subseteq X$ such that $\{N(x): x\in D\}$ ...
1
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0answers
96 views

Category-theoretic characterization of zero-dimensional spaces

Some background: a zero-dimensional space is one admitting a basis of clopen sets, whereas an extremely disconnected space is one where the closures of open sets are open. In the category CHauss of ...
3
votes
0answers
74 views

Adjunctions of uniformly locally connected spaces

A space $X$ is uniformly locally connected (ULC) if there exists an open neighbourhood $U$ of the diagonal $\vartriangle_X$ in $X \times X$ and a homotopy $H: U \times I \to X$ between $\pi_1|U$ and $\...
39
votes
1answer
1k views

Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?

Similarly is the complement of any countable set in $\mathbb R^3$ simply connected? Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...
0
votes
1answer
70 views

Does order-preserving equal continuous? [closed]

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?
1
vote
1answer
71 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 ...
3
votes
1answer
92 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 $P$?
3
votes
1answer
153 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: y\...
12
votes
2answers
409 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 $$X_0\rightarrow X_1\rightarrow\ldots$$ $$Y_0\rightarrow Y_1\rightarrow\ldots$$ $$Z_0\...
3
votes
0answers
103 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 ...
1
vote
1answer
118 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
78 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
169 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 ...
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0answers
180 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 $\beta\...