**6**

votes

**1**answer

314 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?
I refer to the (systematic, formalist) study of the category of sheaves on a site or the study of topology in a ...

**5**

votes

**0**answers

99 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 ...

**3**

votes

**0**answers

89 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 ...

**4**

votes

**1**answer

126 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, ...

**1**

vote

**0**answers

26 views

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

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 ...

**15**

votes

**2**answers

597 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 ...

**7**

votes

**1**answer

260 views

### Topological conditions forcing continuity

Let $X$, $Y$, and $Z$ be topological spaces. Let $f:X \rightarrow Y$. Further assume that for every continuous function $g:Y \rightarrow Z$, $g \circ f$ is continuous.
Question: Under what ...

**9**

votes

**1**answer

184 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 $|\frak T|\le |\Bbb R|$?

**7**

votes

**5**answers

596 views

### Defining a topology in the Power Set

I have the follwing question:
Given a topological space $T$ is possible in general to give a topology to $2^T$ (the power set of $T$) such that this topology in $2^T$ is related to $T$.
If the ...

**15**

votes

**7**answers

869 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 ...

**0**

votes

**1**answer

89 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 ...

**7**

votes

**3**answers

622 views

### Is the reals the smallest connected ordered topological ring?

The real numbers is a locally compact Tychonoff connected complete ordered topological field. I am looking at minimal collections of adjectives that can characterize the reals. The one often used to ...

**6**

votes

**2**answers

292 views

### A continuous notion of realizability

I have been interested in non-classical logics, off and on, for quite a while. This question is probably very basic, and I hope it is not too low-level for MO. My question stems from an attempt to ...

**2**

votes

**0**answers

113 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 ...

**22**

votes

**1**answer

1k views

### Fake versus Exotic

Without recourse to the Disc Theorem (or its progeny), is it true that all known examples of exotic differentiable structures on 4-manifolds would be fake rather than exotic?
Terminology (perhaps ...

**3**

votes

**1**answer

160 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 ...

**20**

votes

**2**answers

841 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 ...

**2**

votes

**1**answer

57 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 $N\setminus O$.
...

**19**

votes

**0**answers

464 views

### Are there “chain complexes” and “homology groups” taking values in pairs of topological spaces?

Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A ...

**1**

vote

**0**answers

139 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 ...

**1**

vote

**0**answers

46 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 ...

**7**

votes

**1**answer

257 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 ...

**20**

votes

**1**answer

684 views

### Computing Self-Intersections with Complex Analysis

It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis:
$$n = \oint_C\frac{dz}{z}.$$
You can also count the number of roots of $f(z) = 0$ inside a close ...

**6**

votes

**0**answers

363 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 ...

**10**

votes

**1**answer

507 views

### Reference request for TQFT, functoriality

I am reading Turaev's blue book Quantum Invariants of Knots and 3-manifolds.
It is difficult for me to understand the proof of Theorem 1.9 in chapter 4, which says;
The function $(M, \partial_{-}M, ...

**6**

votes

**1**answer

160 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 ...

**8**

votes

**0**answers

449 views

### A proof of the gluing axiom of a TQFT

I posted the following question on math stackexchange but I have not received any answer.
So I hope people here can help me.
In the book Lectures on tensor categories and modular functors by Bakalov ...

**2**

votes

**1**answer

85 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 ...

**0**

votes

**1**answer

55 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 ...

**21**

votes

**4**answers

3k views

### Why are the integers with the cofinite topology not path-connected?

An apparently elementary question that bugs me for quite some time:
(1) Why are the integers with the cofinite topology not path-connected?
Recall that the open sets in the cofinite topology on ...

**2**

votes

**0**answers

66 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

263 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 ...

**3**

votes

**0**answers

71 views

### Non Borel Spaces: Gauge Integral

Question
Is there a generalization of the gauge integral to measure spaces that do not necessarily arise out of some topology?
I'm wondering since it seems as the gauge crucially uses ...

**11**

votes

**2**answers

444 views

### Manifolds covered by an n-dimensional torus

It is well-known that classification of manifolds up to homemorphism is, in general, out of question. However, this task is sometimes tractable under some additional assumptions on manifolds one would ...

**2**

votes

**3**answers

206 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 ...

**5**

votes

**0**answers

210 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 ...

**12**

votes

**5**answers

2k views

### Locally compact Hausdorff space that is not normal

What is a good example of a locally compact Hausdorff space that is not normal? It seems to be well-known that not all locally compact Hausdorff spaces are normal (and only a weaker version of ...

**16**

votes

**5**answers

1k views

### A topological concept dual to compactness

We say that a subset A in a topological space X is anti-compact if every covering of A by closed sets has a finite subcover. Clearly if X is Hausdorff then all anti-compact subsets of X are finite. ...

**2**

votes

**1**answer

79 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

**0**answers

167 views

### Number of faces of a polytopal subdivision

Let $\mathcal{P}$ be a (bounded) polytope in $\mathbb{R}^d$ and let $\mathcal{C}$ be a polytopal subdivision of $\mathcal{P}$ [1].
Is there a known tight upper bound in the number of polytopes in ...

**14**

votes

**2**answers

938 views

### Generalization of Darboux's Theorem

Darboux's Theorem. If $f:[a,b]\to\mathbb R$ is differentiable and $f'(a)<\xi<f'(b)$, then there exists a $c\in (a,b)$, such that $\,f'(c)=\xi$.
Does any of the following generalizations
Let ...

**0**

votes

**1**answer

69 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

**20**

votes

**1**answer

775 views

### A function composed with itself produces the identity

Let $B$ be the closed unit ball in $\mathbb R^3$ and $f: B\to B$ continuous, such that $f\circ f$ is the identity (i.e., $f\circ f=\mathbb 1_B$) and $f$ restricted on $\partial B$ is also the identity ...

**2**

votes

**0**answers

54 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 ...

**5**

votes

**2**answers

406 views

### A question about the Stone–Čech compactification of discrete spaces

Let $D(\kappa)$ be the discrete space of cardinality $\kappa$, and $\beta D(\kappa)$ its Stone–Čech compactification.
Is there, for every infinite cardinal $\kappa$, a subset $Y \in [\beta ...

**16**

votes

**2**answers

557 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 ...

**14**

votes

**8**answers

2k views

### References for homotopy colimit

(1) What are some good references for homotopy colimits?
(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...

**12**

votes

**1**answer

235 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 ...

**3**

votes

**1**answer

192 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 ...

**8**

votes

**1**answer

310 views

### Partition $\Bbb{R}$ into a family of sets each one homeomorphic to the Cantor set

It is known that there is no (nontrivial) partition of $\Bbb{R}$ into a countable number of closed set. But is there a partition of $\Bbb{R}$ into sets, each one homeomorphic to the cantor ternary ...