**7**

votes

**1**answer

102 views

### Product of Limit $\sigma$-Algebras

Let $X$ and $Y$ be Polish (i.e. Borel subsets of separable completely metrizable) spaces. For a Polish space $Z$, let $\mathscr{S}(Z)$ denote the limit $\sigma$-algebra on $Z$, i.e. the smallest ...

**0**

votes

**0**answers

51 views

### Reference request: Uniformly totally bounded classes of compact metric spaces are Gromov-Hausdorff precompact

The following Theorem can be found for instance here (Theorem 7.4.15):
Theorem. (author ?) Any uniformly totally bounded class $\mathfrak X$ of compact metric spaces is pre-compact in the ...

**12**

votes

**1**answer

520 views

### Duality between topology and bornology

I want to understand in what sense topology is dual to bornology at a most basic level. Therefore, I rephrased the definition of a bornology in the following way:
Let $X$ be a set and let ...

**38**

votes

**1**answer

983 views

### Does there exist a bijection of $\mathbb{R}^n$ with 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, $f(S)\subset Y$ ...

**30**

votes

**2**answers

3k views

### “Transitivity” of the Stone-Cech compactification

Let $\beta \mathbb{N}$ be the Stone-Cech compactification of the natural numbers $\mathbb{N}$, and let $x, y \in \beta \mathbb{N} \setminus \mathbb{N}$ be two non-principal elements of this ...

**4**

votes

**0**answers

174 views

### Unbounded towers and combinatorial cardinal characteristics of the continuum

Update: Perhaps the question is too difficult. I would appreciate, thus, even just comments or related observations.
This question assumes familiarity with combinatorial cardinal characteristics of ...

**7**

votes

**1**answer

249 views

### Abstract result on partitions of unity?

A motivation: The classical Stone-Weierstrass theorem says that polynomials are dense among continuous functions (say, on the unit interval), while the abstract Stone-Weierstrass theorem (and also the ...

**2**

votes

**2**answers

254 views

### Is it true that all sphere bundles are some double of disk bundle?

Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you ...

**1**

vote

**0**answers

28 views

### Zero-dimensional $F$-space which is not strongly zero-dimensional

Does anyone know of an example of a (Tychonoff) $F$-space which is zero-dimensional but not strongly zero-dimensional?
By an $F$-space we mean every cozeroset is $C^*$-embedded.
By zero-dimensional ...

**1**

vote

**0**answers

45 views

### Dual equivalence for multioperators

This is a reference request question. But let's start with a few definitions.
Let $L$ and $M$ be two bounded lattices. A multioperator $p$ for $L$ and $M$ is an application $$p : L \to Ft(M)^{op}$$ ...

**3**

votes

**2**answers

451 views

### SU(2) representations of alternating knot groups

Suppose that $K$ is an $\textit{alternating}$ knot in $S^3$, and let $R_0$ be the space of homomorphisms from $\pi_1(S^3 - K)\to SU(2)$ which send meridians to trace free matrices. Denote the subset ...

**28**

votes

**1**answer

690 views

### What is the meaning of this analogy between lattices and topological spaces?

Let me add one more edit to help explain why this is a serious question. Theorem 5 below is a sort of lattice version of Urysohn's lemma, and it has essentially the same proof. Theorem 6, the famous ...

**5**

votes

**1**answer

287 views

### When is there an unbounded tower in $[\mathbb{N}]^\infty$?

(Edit: I'm splitting the question, leaving here only what is answered by Ashutosh, and moving the rest to another question.)
This question assumes familiarity with combinatorial cardinal ...

**4**

votes

**2**answers

487 views

### Some examples of clean topological spaces

I asked this question at MSE but I did not received any answer, so I repeat it here at MO:
What is an example of a Hausdorff topological space $X$, not a singleton, such that the ring ...

**6**

votes

**0**answers

115 views

### Is every Polish ring topology on $\mathbb{C}$ defined by an absolute value?

There is a unique up to isomorphism algebraically closed field of characteristic 0 and cardinality of the continuum. Let's call it $K$.
We usually call it $\mathbb{C}$, but by this we impose a ...

**30**

votes

**4**answers

1k views

### Connectedness in the language of path-connectedness

Is there a topological space $(C,\tau_C)$ and two points $c_0\neq c_1\in C$ such that the following holds?
A space $(X,\tau)$ is connected if and only if for all $x,y\in X$ there is a ...

**0**

votes

**0**answers

74 views

### Right split for homomorphism onto $S_\infty$

Let $G$ be a closed subgroup of $S_\infty$ and let $f:G\rightarrow S_\infty$ be a continuous surjective homomorphism. Under which conditions $f$ has a right split, i.e. there exists some ...

**1**

vote

**0**answers

109 views

### $\mathbb E$-descent maps in topological spaces in terms of different sites?

The paper Facets of Descent I by Janelidze and Tholen defines $\mathbb E$-descent maps as those for which $\Phi^p:\mathbb EB\longrightarrow \mathsf{Des}_\mathbb{E}(p)$ is an equivalence of categories.
...

**10**

votes

**1**answer

181 views

### Open (resp., closed) balls homeomorphic to open (resp., closed) discs on the plane

Let $\Sigma$ be a compact (smooth) surface, with a geodesic metric $d$ (compatible with the topology of $\Sigma$).
Let $x \in \Sigma$, and suppose you have the following: for every $r<1$,
the open ...

**19**

votes

**12**answers

2k views

### Classic applications of Baire category theorem

I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...

**1**

vote

**0**answers

60 views

### The role of absolute continuity in stochastic ordering defined over sets of probability distributions

This question is about a claim given in this paper (page 261, the remark), but without any proof.
It simply says that if two sets of probability distributions, $\mathscr{P}_0$ and $\mathscr{P}_1$ ...

**7**

votes

**1**answer

203 views

### What is the height (or depth) of $[\mathbb{N}]^\infty$?

(This question assumes familiarity with combinatorial cardinal characteristics of the continnum.)
Let $[\mathbb{N}]^\infty$ be the family of infinite subsets of $\mathbb{N}$,
partially ordered by ...

**8**

votes

**2**answers

425 views

### Topological characterization of injective metric spaces

Let $\ (X\ d)\ \,(Y\ \delta)\ $ be arbitrary metric spaces. A function $\ f:X\rightarrow Y\ $ is called a metric map (with respect to the given metrics $\ d\ \delta$) $\ \Leftarrow:\Rightarrow\ ...

**-4**

votes

**1**answer

405 views

### For every proximity, does there exist a uniformity which generates this proximity?

For every proximity, does there exist a uniformity which generates this proximity?
This question may be generalized for different generalizations of proximities and uniformities. In fact I need it ...

**2**

votes

**1**answer

145 views

### Relation between two different definitions for relative sequential compactness

Building upon this question in Math.SE, I think the following might be rather of interest for MO.
In the literature on measure theory, probability and functional analysis the definition of a subset ...

**1**

vote

**3**answers

269 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 ...

**15**

votes

**3**answers

989 views

### Topological spaces whose continuous image is always closed

If $X$ a topological space one says that $X$ is universally closed if for every Hausdorff space $Y$ and every (continuous) map $f:X\rightarrow Y$, the image of $X$ is a closed subset of $Y$.
It is ...

**2**

votes

**1**answer

105 views

### Surniversal spaces

Basic background
On one hand there is a complete result: $\,\ $for every non-negative integer $n$ there exists an $n$-dimensional compact metric space $M^n$ such that it contains a homeomorphic ...

**2**

votes

**1**answer

222 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 ...

**0**

votes

**0**answers

31 views

### Free cocompact action of discrete group gives a covering map [migrated]

I'd like to find a short proof of the following seemingly basic fact encountered on the second page of Atiyah's paper "Elliptic operators, discrete groups, and von Neumann algebras." ...

**7**

votes

**2**answers

331 views

### Link between the hairy ball theorem and the fundamental theorem of algebra

I read in the book "Concepts of modern mathematics" by Ian Stewart that it was possible to proof the fundamental theorem of algebra using the hairy ball theorem (complete reference to the page is in ...

**4**

votes

**1**answer

107 views

### Minimal zero-dimensional Hausdorff spaces

A topological space $(X,\tau)$ is said to be zero-dimensional Hausdorff (zdH) if for $x\neq y\in X$ there is $C\subseteq X$ clopen (closed and open) such that $x\in C$, but $y\notin C$.
We say a zdH ...

**2**

votes

**1**answer

67 views

### Topology with no direct lower neighbor

Given any poset $(P,\leq)$ and $x, y\in P$ we set $[x,y] = \{p\in P: x\leq p \leq y\}$. For any set $X$, let $\text{Top}(X)$ denote the set of topologies on $X$. The set $\text{Top}(X)$ is a complete ...

**9**

votes

**2**answers

341 views

### A question about the dispersion points of connected metric spaces

Let $C$ be an infinite, separable and connected metric space. If $C$ becomes totally disconnected when one of its points $p\in C$ is removed, does every closed ball of $C$ with
positive radius and ...

**0**

votes

**0**answers

41 views

### Fixed point shape property

Question: Provide (or prove that it's not possible) a metric compact space which has the fixed point property but not the fixed point shape property.
Here is the definition of f.p.s.p.("map" means ...

**4**

votes

**0**answers

36 views

### Intuition for universal quotient maps

The universal quotient maps are precisely the descent morphisms in the category of topological spaces. In some papers of Janelidze, Tholen, Sobral, and Reiterman (see for instance Reiterman-Tholen), ...

**8**

votes

**1**answer

169 views

### Embedding abelian cancellative Hausdorff topological semigroups into abelian Hausdorff topological groups

An abelian cancellative semigroup embeds (via a semigroup monomorphism) into an abelian group. What about an abelian cancellative Hausdorff topological semigroup that does not embed (via a ...

**1**

vote

**1**answer

189 views

### Extension of continuous and smooth functions

Let us consider any subset $U \subset \mathbb{R}^{n}$. By definition, a function $f: U \rightarrow \mathbb{R}^m$ is smooth if, for every $x \in U$, there exist an open neighbourhood $\Omega_{x}$ of ...

**4**

votes

**3**answers

155 views

### End points of continua

Whyburn (1942) defined an end point $x$ of a continuum $X$ to be any point having arbitrarily small neighborhoods each of whose boundaries contains a single point. Thus, he defines an end point ...

**21**

votes

**1**answer

305 views

### Is the normal bundle of a torus trivial?

Question:
Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial?
What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the ...

**16**

votes

**2**answers

1k views

### complement of a totally disconnected closed set in the plane

While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected ...

**1**

vote

**1**answer

172 views

### Applications of topology to discrete dynamical systems?

I'd like to know some of the applications of topology to discrete dynamics. By discrete dynamics I loosely mean studying maps between discrete sets.
I mean cases where adding a topology to the sets ...

**3**

votes

**1**answer

180 views

### Fréchet L-Spaces

According to the paper The emergence of open sets, closed sets, and limit points
in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...

**2**

votes

**1**answer

238 views

### Local “pathologies” in spaces arising naturally in algebraic topology

I have been thinking about methods for constructing continuous paths locally in a space. These paths have domain the unit interval and map into "small" neighborhoods of points in a space. Moreover ...

**1**

vote

**1**answer

87 views

### Lower neighbors in the lattice of topologies

Given any poset $(P,\leq)$ and $x, y\in P$ we set $[x,y) = \{p\in P: x\leq p < y\}$, and $(x,y]$ is defined in an analogous manner. For any set $X$, let $\text{Top}(X)$ denote the set of topologies ...

**14**

votes

**0**answers

358 views

### What is the Cantor-Bendixson rank of the space of first order theories?

Let $L$ be the language $\{R\}$ containing a single binary relation symbol. Consider the space $S_0(L)$ of complete, first-order $L$-theories. This is a seperable, compact Hausdorff space; what is its ...

**3**

votes

**1**answer

79 views

### “Discrete jumps” in the collection of all topologies on a set $X$

Given any poset $(P,\leq)$ and $x, y\in P$ we set $[x,y] = \{p\in P: x\leq p \leq y\}$. For any set $X$, let $\text{Top}(X)$ denote the set of topologies on $X$. The set $\text{Top}(X)$ is a complete ...

**1**

vote

**2**answers

116 views

### Smooth, irreducible surface with real part containing two projective planes

Let $X$ be a smooth and irreducible projective variety over $\mathbb{R}$ of dimension two. I am looking for an instance of such a variety where two distinct connected components of $X(\mathbb{R})$ are ...

**1**

vote

**0**answers

116 views

### Is every path connected space continuously path connected

Recall a topological space $X$ is path connected if for all $x,y \in X$ there is a continuous function $f\colon [0,1] \to X$ such that $f(0)=x$ and $f(1)=y$.
Say that $X$ is continuously path ...

**1**

vote

**1**answer

97 views

### Intersections of families of open sets ordered by well-inside relation in Euclidean space

Let $\langle\mathbf{R}^n,\mathscr{O}\rangle$ be the $n$-dimensional Euclidean space. Define $\mathbf{Q}\subseteq\mathcal{P}(\mathscr{O})$ to consist of all sets $\mathsf{Q}$ which simultanously ...