**3**

votes

**1**answer

110 views

### Lipschitz function with somewhere dense image

Let $Q=[-1,1]^2$ denote the unit square and let $f:Q\to Q$ be a Lipschitz function such that for any ball $B(a,r)\subset Q$ with radius $r$, the width of the image $f(B(a,r))$ is at least $cr$ for ...

**7**

votes

**3**answers

380 views

### Binary relations as the topological closure of the diagonal

If $(X,\tau)$ is a topological space, we can consider the product topology on $X\times X$ and take the closure of the diagonal $\Delta_X = \{(x,x): x\in X\}$, which we denote by ...

**7**

votes

**1**answer

427 views

### Group structure on an arbitrary completely regular topological space that makes $(x,y)\mapsto xy^{-1}$ continuous at $(1,1)$

Let $(G,\mathcal T)$ be a completely regular topological space. Is there a group structure on $G$ such that the function
$$f:G\times G\to G$$
$$f(x,y)=xy^{-1}$$
is continuous at $(1,1)$?

**6**

votes

**0**answers

158 views

### A compactification of the non-negative rationals with the discrete topology

Let $S$ be the set of non-negative rational numbers. (If it makes any difference, feel free to take the non-negative dyadic rationals instead.) Let $B=\ell_\infty(S)$; as a ${\rm C}^*$-algebra this is ...

**10**

votes

**1**answer

725 views

### Reference request: Book of topology from “Topos” point of view

Question: Is there any book of topology in the modern language of topos theory?
Motivation:
In "Sheaves in Geometry and Logic" Mac Lane and Moerdijk say: "For Grothendieck, topology became the ...

**1**

vote

**0**answers

34 views

### The topology of complete minimal surfaces of finite total Gaussian curvature [closed]

Suppose that M is
a complete minimal surface with finite total curvature. If M is embedded in $\mathbf{R}^3$, then
we observe that M viewed from infinity looks like a plane passing through the ...

**11**

votes

**1**answer

259 views

### Maximum length of a chain of topologies on $\Bbb R$

Let $\frak T$ be a totally ordered set of topologies on $\Bbb R$.
Is $|\frak T|\le |\Bbb R|$?

**3**

votes

**0**answers

138 views

### $S^{3}$-valued harmonic analysis

Edit:
Note that $S^{3}$ with the quaternion operation is a group. For a locally compact Abelian group $\Gamma$ we consider
$$\tilde{\tilde{\Gamma}}=\{\phi:\Gamma \to S^{3} \mid ...

**2**

votes

**1**answer

71 views

### Separating Differences of Open Sets

Has anyone ever considered something like the following separation axiom?
$(*)$ For any pair of open sets $O$ and $N$, there exist disjoint open sets containing $O\setminus N$ and $N\setminus O$.
...

**21**

votes

**2**answers

1k views

### A question about “Zariski dense” arguments

This question is a little basic, but I think it is consistent with the goals of MO.
My question is about a certain type of argument in algebraic geometry which exploits the abundance of dense sets ...

**4**

votes

**1**answer

173 views

### Are infinite groups “locally topologizable”?

Does every infinite group admit a Hausdorff topology such that the multiplication and inverse are continuous at $1$ but $1$ is not an isolated point?
The question is inspired by and related to ...

**25**

votes

**9**answers

1k views

### Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous? In topology, category theory or in analysis? Is it possible, canonical, useful?
I have a vague idea of the possibility of ...

**2**

votes

**0**answers

83 views

### Ostaszewski space's construction Lemma

I'm studying the Ostaszewski's article "On Countably Compact, Perfectly Normal Spaces". I'll add some context. Lemma 1.2 says the following:
Let $X$ be a locally compact, zero-dimensional and ...

**1**

vote

**0**answers

169 views

### Continuous dependence of the roots of a polynomial on its coefficients

In their article "The roots of a polynomial vary continuously as a function of the coefficients" Gary Harris and Clyde Martin give a topological proof of the well-known theorem that the roots of a ...

**4**

votes

**1**answer

141 views

### Is the absolute of a compact space the projective limit of the Stone-Čech compactifications of its open dense subsets?

Is the following statement true, and if it is, does someone have a reference?
Let $X$ be a compact (i.e., compact and Hausdorff) topological space. Then the Gleason space (=Iliadis absolute, ...

**9**

votes

**0**answers

423 views

### A group allowing exactly 7 group topologies

Is there a group $G$ allowing exactly 7 group topologies on $G$: $\mathcal T_{trivial}, \mathcal T_{discrete}, \mathcal T_1, \mathcal T_2,\mathcal T_3,\mathcal T_4, \mathcal T_5$ with
$$\mathcal ...

**8**

votes

**2**answers

340 views

### Ultracoproducts and Cartesian products

Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$.
As a C$^*$-algebraist, I ...

**2**

votes

**1**answer

100 views

### A Property of Baire Spaces

I posed this on August 8 on math.stackexchange, but there has been no response.
Let $X$ be a topological space. I define $X$ to have Property A provided that every closed meager subset of $X$ is ...

**8**

votes

**2**answers

253 views

### Finiteness as a motivation for compactness

Another history question, and I am not sure if I will get any answers. (If anyone knows of a good history of math list to use for this question I would be happy for any tips. The one I used to post to ...

**1**

vote

**1**answer

84 views

### Continuous real function on germs

Let $C_0^{m,n}$ be the space of germs of continuous maps from $\mathbb{R}^m$ to $\mathbb{R}^n$, located at $0\in\mathbb{R}^m$, with the usual inductive limit topology. One can also consider ...

**3**

votes

**0**answers

138 views

### Decreasing sequence of closed convex sets in a Banach space

Let $(C_n)$ be a decreasing sequence of closed convex subsets in a Banach space $(E,\Vert \cdot \Vert)$. The question I have is about the content of $C=\bigcap_{n=0}^\infty C_n$.
If the $C_n$ are ...

**3**

votes

**0**answers

138 views

### History of limit point compact -/-> compact example

A standard example in elementary topology (e.g. Munkres) of a space which is limit-point compact (every infinite subset of the space has a limit point) but not compact is the minimal uncountable ...

**5**

votes

**2**answers

306 views

### non-Borel set which intersects every compact in a Borel set

I remember hearing some time ago that there is a locally compact Hausdorff space $X$ and a non-Borel subset $E$ which intersects every compact set in a Borel set. (This would contradict Lemma 13.9 of ...

**5**

votes

**0**answers

222 views

### Quotienting disk inside sphere result in sphere

Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let
$q: D^k \to D^r$ be a map and $r \leq k$. Let
$$W = S^k \sqcup D^r/\sim$$
where ...

**2**

votes

**1**answer

92 views

### Going Back-and-Forth Between Different Expressions/“Representations” for Open Books.

I am trying to have a better understanding of how one goes , "travels" between the different formats/layouts of open books for a fixed given 3-manifold M; between the abstract type and the "actual" ...

**1**

vote

**2**answers

143 views

### Why the intersection of a scott open (or \w the relatively compactness property) filter on a topology of a sober (and 2nd countable) space is compact?

Definitions and notations.
Let $\mathcal{P}(X)$ the power set of $X$.
Let $\tau_X\subseteq\mathcal{P}(X)$ a topology on X.
We call $A$ irreducible if every time $A=B\cup C$ with $B,C$ closed set ...

**0**

votes

**1**answer

74 views

### Topology : Study on Separation Properties [closed]

I want an example of a completely regular space which is not a normal space. I have tried a lot but am unable to construct any example

**2**

votes

**0**answers

64 views

### d-refining covering of normal space

If $X$ is normal, it is well known that for any open-covering $(U_i)$ of $X$, there exist closed subspaces $F_i$ and $G_i$ and an open subspaces $O_i$ such that $$F_i\subset O_i\subset G_i\subset ...

**3**

votes

**1**answer

254 views

### Quotients of standard Borel spaces

Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation ...

**17**

votes

**1**answer

611 views

### Topological transversality

Warmup question:
Let us say that two continuous functions $f,g:[0,1]\to \mathbb R$ are topologically transverse if their difference $f-g$ has only finitely many zeros, and each zero separates an ...

**5**

votes

**0**answers

64 views

### Separation of topological group elements by invariant neighbourhooods

Let $G$ be a topological group that is Hausdorff, that is, for every pair $(g,h)$ of distinct elements of $G$, there exist disjoint open sets $U_g$ and $U_h$ such that $g \in U_g$ and $h \in U_h$.
...

**4**

votes

**2**answers

159 views

### Are Hausdorff compactifications of a Tychonoff space $X$ in one-to-one correspondence with completely regular subalgebras of $BC(X)$?

Let $X$ be a completely regular (Tychonoff) topological space. It is known that if $\mathscr F\subseteq C(X,[0,1])$ separates points and closed sets (that is, for every closed set $E\subseteq X$ and ...

**5**

votes

**1**answer

127 views

### Is an extension of compact Hausdorff topological groups compact?

Let $1 \rightarrow A \xrightarrow{a} B \xrightarrow{c} C \rightarrow 1$ be a short exact sequence of topological groups (i.e., all maps are continuous, $A = \mathrm{Ker}(c)$, and $C = ...

**18**

votes

**2**answers

742 views

### Which sets occur as boundaries of other sets in topological spaces?

This question was originally asked on MathStackExchange and is migrated here with opinion from MO meta. I am integrating the inputs from users Daniel Fischer and Emil Jerabek there into this post.
...

**5**

votes

**1**answer

151 views

### A realcompact analogue of the Baire category theorem

Let $\frak{m}$ be the least measurable cardinal. A space $X$ is realcompact if it is homeomorphic to a closed subset of some product $\mathbb{R}^I$. Let $X$ be realcompact with $P_\frak{m}$ topology, ...

**9**

votes

**2**answers

201 views

### A differentiable one-parameter family of codimension 2 subspaces of $\mathbb{C}^n$ cannot fill $\mathbb{C}^n$, right?

Suppose that $P(t)$ is a one-parameter family of rank 2 self-adjoint projections on $\mathbb{C}^n$ that vary analytically in the real parameter $t \in [0,1]$. I claim that there must exist a vector $x ...

**12**

votes

**1**answer

267 views

### A generalization of the Arhangelskii Theorem

Arhangeleskii's Theorem states the following
For any Hausdorff topological space $X$,
$$
|X|\leq2^{\chi(X)L(X)}
$$
where $\chi(X)$ is the character of $X$ and $L(X)$ is the Lindelöf degree of ...

**0**

votes

**0**answers

62 views

### Can a “weak” topological space be a Moore space?

Let B be a reflexive and infinite dimensional real Banach space-which could be Hilbert space l^2- and let B be endowed with the weak topology. Although this topology is regular and Hausdorff, it is ...

**3**

votes

**3**answers

389 views

### “countable” topology

Given universal set $U$. Is there any name of the collection of subsets of $U$ (call them quasi-open) satisfying the following axioms:
i) $\emptyset$ and $U$ are quasi-open;
ii) finite intersections ...

**2**

votes

**0**answers

44 views

### Relative isotopy of simple curves in a disk

Consider the closed two dimensional disk and two fixed points $A$ and $B$ on the boundary. I am looking for a reference for the following fact: a simple topological curve with end points $A$ and $B$ ...

**2**

votes

**0**answers

150 views

### Homeomorphism of compact Hausdorff spaces

(Note: I asked this question at MSE over a day ago and received no answer, so I'm now reposting it here. Link: http://math.stackexchange.com/questions/853500/homeomorphism-of-compact-hausdorff-spaces)
...

**2**

votes

**0**answers

90 views

### Berkovich Analytification of the transseries

I am looking for references to articles about the following subjects:
Connections from the field of (real) transseries to the field of surreal numbers (mentioned very briefly in the introduction of ...

**6**

votes

**1**answer

228 views

### Homeomorphism between derived sets implies homeomorphism

(Note: I asked this question at MSE days ago and received no answer, so I'm now reposting it here.)
I want to prove the following statement:
Let $K_1$ and $K_2$ be two countable, compact sets of ...

**2**

votes

**1**answer

99 views

### open subsets of boundary [closed]

Let $\bar{X}$ be a Hausdorff and compact topological space. Suppose that $X$ is an open and dense subset of $\bar{X}$. Let $\nu X=\bar{X}\setminus X$ and assume that $U\subseteq \nu X$ is an open ...

**3**

votes

**0**answers

154 views

### Compact set covered by two opens

The following lemma about locally compact (but not necessarily Hausdorff) spaces or continuous lattices appears frequently but without citation. It is easy to prove but important in proofs.
If a ...

**4**

votes

**1**answer

194 views

### A question on $Z^{*}$ algebras

A $Z^{*}$ algebra is a $C^{*}$ algebra which satisfies each of the following equivalent conditions:
All elements of $A$ are left zero divisor.
All elements are right zero divisor.
All elements ...

**14**

votes

**0**answers

156 views

### Finite union of closed convex sets is triangulable?

I posted this question on Stackoverflow, but didn't get an answer.
Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable, that is, ...

**2**

votes

**3**answers

241 views

### Compact, densely ordered spaces

During my work with order preserving homeomorphisms, I got interested in the double arrow space and, subsequently, in the lexicographic square.
I would really like to find examples of spaces like ...

**1**

vote

**1**answer

168 views

### Sober topological subspace

Assume $X$ to be a Notherian topological space such that any irreducible closed
subset has a unique generic point. Consider $Y\subseteq X$ as a topological space with the induced topology from $X$. Is ...

**4**

votes

**2**answers

295 views

### If $E$ maps onto a contractible space with contractible fibers, must $E$ be contractible?

Let $p\colon E\to C$ be a continuous, surjective map between topological spaces with $C$ contractible. Suppose that $p^{-1}(c)$ is contractible for each $c\in C$. Is it true that $E$ is weakly ...