Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

learn more… | top users | synonyms (2)

33
votes
0answers
868 views

Differential forms on fuzzy manifolds

This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me. Question In the usual setting of open subsets of ...
26
votes
0answers
712 views
+100

Is there a homology theory that counts connected components of a space?

It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$. I have recently learned that for Čech homology the ...
23
votes
0answers
308 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, $f(S)\subset Y$ ...
21
votes
0answers
437 views

In $n>5$, topology = algebra

During the study of the surgery theory I faced following sentence: Surgery theory works best for $n > 5$, when "topology = algebra". I don't know what is the meaning of topology=algebra. ...
16
votes
0answers
724 views

Lifting local compactness in covering spaces

Since the total space of a cover is locally homeomorphic to the base space, local topological properties (like local (path) connectedness, T1 etc.) lift from the base space to the total space. The ...
14
votes
0answers
288 views

On distributivity of lattice of group topologies

Let $\frak L$ be the set of all topologies $\mathcal T$ on $\Bbb Q$ (the additive group of all rational numbers) such that $(\Bbb Q,\mathcal T)$ is a topological group. Then $(\frak L,\subseteq)$ is a ...
13
votes
0answers
137 views

Strengthening the intermediate value theorem to an “intermediate component theorem”

Let $f$ be a continuous function on a closed Euclidean ball (in dimension $\ge 2$) that is negative in the center of the ball and positive on its boundary. On any path from the center of the ball to ...
13
votes
0answers
551 views

Understanding Alexandroff compactification

Is the Alexandroff one-point compactification of a locally compact Hausdorff space ($\mathbf{LCHaus}$) a functor to the category of compact Hausdorff spaces ($\mathbf{CHaus}$)? It seems to me that one ...
10
votes
0answers
79 views

Axiomatizing topology through continuous maps

Suppose we have some topological space $X$ and we somehow forgot about the topology. A friend of ours knows the topology and offers to tell us for any map $X\to Y$ into any topological space $Y$ ...
10
votes
0answers
216 views

Kernels in $\mathbf{Top}$

There is a following well-known theorem for abelian categories (at least the ones I know, Ab, $R$-mod and so on... not so familiar with categorical language to be honest) which states the following : ...
10
votes
0answers
181 views

Restrictions of null/meager ideal

Let I denote the null/meager ideal on reals. Is it consistent that for any pair of non null/meager sets A and B, there is a null/meager preserving bijection between A and B? In particular, is this ...
10
votes
0answers
201 views

What properties are preserved under a measurable mapping?

Although in an abstract category the morphisms are not explicitly defined, in a concrete example (model theory?), morphisms are (always/usually?) mappings that preserve some properties. In the ...
10
votes
0answers
513 views

A fiber bundle over Euclidean space is trivial.

What's the easiest way to see this? The only thing I could think to do was try to patch together trivializations. I couldn't find a way to make that work. Thank you! edit: For the record, here's why ...
9
votes
0answers
174 views

On infinite groups admitting finitely many group topologies

It has been proved there is an infinite group which admits exactly two group topologies [1]. For which $n$, is there an infinite group $G$ which admits exactly $n$ group topologies ordered linearly ...
9
votes
0answers
75 views

What's the minimal structure needed to define a notion of derivative?

I know that, for example, to define a limit all you need is the notion of "closeness" generated by a topology; and to define an integral you need a measure function and a sigma-algebra on which it is ...
9
votes
0answers
215 views

How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
9
votes
0answers
138 views

When is $\{ x | f(x) \le 0\}$ path-connected?

I'm trying to determine the conditions on $f : \mathbb{R}^n_{\ge 0} \to \mathbb{R^n}$ under which $\{ x | f(x) \le 0\}$ is path-connected. We can assume that $f$ is continuous and concave (i.e. for ...
9
votes
0answers
160 views

Open map which “almost fixes” the boundary of an open ball

We have a continuous function $f:\bar{B}\to\mathbb{R}^n$, where $\bar{B}=\{x\in\mathbb{R}^n:\|x\|\le 1\}$, such that if $\|x\|=1$ then $\|f(x)-x\|<\epsilon$, for a fixed $\epsilon\in(0,1)$. We have ...
8
votes
0answers
264 views

A few standard results (on metrizability and relative separation strength) without choice?

I've been going back over some results from Munkres's Topology, and I'm curious about some things.... I know that Choice principles have some connection to the separation axioms (in ZF, at ...
8
votes
0answers
165 views

Existence of a map in a Hilbert space

Let $H$ be an infinite-dimensional Hilbert space, $B$ be its unit ball: $B=\{x\in H: \, \|x\|\leq 1\}$. Does there exist a continuous map $f:H\to H$ such that $f(f(x))=x$ $\forall x\in H$, $f$ has no ...
7
votes
0answers
123 views

Why use the Lefschetz Zeta function?

Given a compact, triangulable space $X$ and a continuous function $f: X\rightarrow X$, then we define the Lefschetz number $\Lambda_{f}$ by $$\Lambda_{f} = \sum_{k\geq0}(-1)^{k}Tr(f_{\ast}\vert ...
7
votes
0answers
97 views

Is every connected subset of the Sierpiński triangle arcwise connected?

I think this should be true. If it's indeed the case, it seems like this should be a known result, so references are welcome. I managed to prove that (assuming $S$ is the connected subset) $S$ ...
7
votes
0answers
91 views

Flabby sheaf comonad

If one Googles sufficiently hard one finds the statement that Roger Godement gives the first example of a comonad, used to compute flabby resolutions of sheaves, in his monograph "Topologie algébrique ...
7
votes
0answers
87 views

“Implicit” subcover

I'm learning about compactness in topological spaces and I wonder if there is an example of a compact topological space $X$ and an open cover $\{U_\alpha\}$ of $X$ for which we can't show explictly a ...
7
votes
0answers
257 views

Fixed point of a shrinking map. Proof.

Could you please verify my proof? Let $f:X \to X$ be a shrinking map on a compact metric space $(X,d)$. In other words: $d(f(x),f(y)) < d(x,y)$ for all $x,y \in X$. Prove: $f$ has a unique ...
7
votes
0answers
239 views

Is there some elementary proof of invariance of domain?

Invariance of domain at least in statement seems a simple result. I mean, the first time I saw the statement I thought: "the proof can't be that bad", but when I searched for it I saw that it needs ...
7
votes
0answers
89 views

Prove $\mathbb{R}^3$ is not the product of two identical topological spaces

I can only prove this for $\mathbb{R}$: If $\mathbb{R}\cong T\times T$, then $T$ embeds in $\mathbb{R}$ as a closed subspace (e.g. $T\times pt$). Since $\mathbb{R}$ is connected, so is $T$. So $T$ ...
7
votes
0answers
136 views

Urysohn's Lemma needn't hold in the absence of choice. Alternate terminology for inequivalent definitions of “normal” spaces?

A topological space $\langle X,\tau\rangle$ is said to be normal if any two disjoint closed subsets are separated by open sets, meaning that for disjoint $E,F\subseteq X$ with $X\setminus E,X\setminus ...
7
votes
0answers
290 views

compact-open metrizability

Given topological spaces $X$ and $Y$ the set $C(X,Y)$ of all continuous functions $f:X\to Y$ becomes a topological space with the compact-open topology (that is the topology generated by the sets ...
7
votes
0answers
112 views

Continuous choice of basis for subspaces

Consider the flag variety (or flag manifold, depending on who you are) $V=\mathrm {Fl} (3,\mathbb C)$ of complete flags of subspaces of $\mathbb C^3$. That is, an element of M is a tuple (L , P) ...
7
votes
0answers
231 views

Is there an abelian cat of topological groups?

There are lots of reasons why the category of topological abelian groups (i.e. internal abelian groups in $\bf Top$) is not an abelian category. So i'm wondering: Is there a "suitably well ...
6
votes
0answers
83 views

How to master general topology for analysis?

I started learning topology long ago. I first exposed myself to metric topology in Baby Rudin and Munkres Topology 2nd ed. Part I. Munkres is my most revisited book ever since. The first big ...
6
votes
0answers
152 views

Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant with ...
6
votes
0answers
111 views

Is there such a thing as 'overtification' (dual to compactification)?

The dual to the notion of compactness for spaces is overtness. This duality is not manifest in the category of spaces but rather in the quantifiers used to define these notions. Is there a process ...
6
votes
0answers
73 views

How many sets can you get by taking closures, complements, and intersections?

The Kuratowski closure-complement problem yields 14 sets which can be formed by taking the closure and the complement of a single set. But if I want to also include such sets as the frontier or ...
6
votes
0answers
336 views

Questions related to intersections of open sets and Baire spaces

EDIT: I have reposted this question on MathOverflow. (The version posted there is more concise, with some details omitted. I have also added a question about pseudobases with similar property.) MO ...
6
votes
0answers
156 views

Point set topology from an algebraic perspective?

I got this idea of viewing a topology as an operation on a ring of sets. Let $\mathcal R = (\mathcal P(X), \cap, \triangle)$ be a ring of sets. ($\triangle$ is the symmetric difference operation and ...
6
votes
0answers
393 views

Is the boundary of a connected set connected?

SOLVED - check out the comments or directly this answer. I was wondering if the boundary of an unbounded component $C$ of the complement of a bounded connected open set $U$ must be connected (in ...
6
votes
0answers
169 views

Orbit space of a free, proper G-action principal bundle

Let $G$ be a topological group and let $r \colon E \times G \to E$ be a continuous right-action on a topological space $X$. If $p\colon E \to B$ is a continuous map into a topological space $B$ such ...
6
votes
0answers
113 views

Reference request - the topology generated by upward and downward closures of antichains.

By definition, the order topology on a totally ordered set $T$ is the coarsest topology such that every subset of $T$ that can be expressed as the upward or downward closure of a singleton is closed. ...
6
votes
0answers
295 views

Homology of Compact Manifolds

Perhaps this is obvious and I am overlooking something, but why are the homology groups of compact manifolds finitely generated?
6
votes
0answers
134 views

Characterization of open maps

I'm looking for different but equivalent definitions of the concept of open map. So Let $X,Y$ be topological spaces and $f:X\longrightarrow Y$ a function, not assumed to be continuous. I conjectured ...
6
votes
0answers
224 views

Understanding the roots of homomorphism and homeomorphism

I understand the formal (i.e. mathematical) definitions of the terms "homomorphism" and "homeomorphism" as they relate to functions, but I am curious as to the origin of these terms. I don't know ...
6
votes
0answers
436 views

The Number of Topologies on a Finite Set

I would like to know if there is like a magical formula to know how many topologies exist on a finite set For example for $X = \{ a, b, c \}$ I found $29$, but I dont know if there are more or how to ...
6
votes
0answers
90 views

“Cutting out” loops from paths

Given a topological space $X$ and a path $p : [0, 1] \to X$ such that $p(0) \neq p(1)$, does there always exist an injective path $p' : [0,1] \to \operatorname{im}p$ such that $p'(0) = p(1)$ and ...
6
votes
0answers
156 views

What are 'weak' forms of Urysohn's lemma, which do not require choice?

Reference: http://web.mat.bham.ac.uk/C.Good/research/pdfs/horror.pdf It is well known that the original proof of Urysohn's lemma uses a choice principle. (DC) What are weak forms of the Urysohn's ...
6
votes
0answers
103 views

Unicoherence of non-euclidean spaces

My question concerns the notion of unicoherence, which is a property that a topological space may or may not have. The definition (from Wikipedia) is: "A topological space $X$ is said to be ...
6
votes
0answers
121 views

Piecewise Affine Bijections of $\mathbb{R}^n$

I have a min-max function $f:\mathbb{R}^n\to\mathbb{R}^n$ of the form $$f(x) = \min_{i=1,\dots,n}\max_{j=1,\dots,n}(\alpha_{ij}^Tx + \beta_{ij})\quad\text{where each } \alpha_{ij}\in ...
6
votes
0answers
379 views

Homotopy extension property vs. good pairs

I'm taking a course that uses the book "algebraic topology" by Allen Hatcher. I this book there are two different ways in which a pair (X,A) of a topological space X and a subspace A can be nice: They ...
5
votes
0answers
73 views

“Wheel Theory”, Extended Reals, Limits, and “Nullity”: Can DNE limits be made to equal the element “$0/0$”?

"Wheels" are a little-known kind of algebraic structure: They modify the concept of a field or a ring in such a way that division by any element is possible, including division by zero, while also ...