# Questions tagged [general-topology]

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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

8,965 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
19k 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, $f(S)\subset Y$ ...
8k views

464 views

### What is the structure preserved by strong equivalence of metrics?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are (topologically) equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
611 views

### Exercise in Engelking's book regarding a disconnected space.

This is related to 6.3.24 in Engelking's Topology book. The Hilbert space $H$ is the set of sequences $(x_i) \in \mathbb R ^\omega$ such that $\|x\|=\sum _{i=1} ^\infty x_i ^2<\infty$, with ...
254 views

### Finite-Dimensional Homogeneous Contractible Spaces

Suppose that $X \subset \mathbb{R}^n$ is compact, homogeneous and contractible (and thus connected). Does $X$ have to be a point? I couldn't think of a non-trivial example, and there isn't a ...
498 views

### Stokes' Theorem general case

With the following lemma : Lemma : Let $f_{+},f_{-} : U \longmapsto \mathbb{R}$ be $C^{1}$ maps with $f_{-} \leq 0 \leq f_{+}$ with $U \subset \mathbb{R}^{n}$ open and bounded with $C^{1}$ boundary. ...
284 views

### Is there a topological space of cardinality $\kappa$ containing all metric spaces of cardinality $\kappa$?

It is known that all the countable metrizable spaces embed in $\mathbb{Q}$. Then a natural question is: are there an uncountable cardinal $\kappa$ and a topological space $X$ of cardinality $\kappa$ ...
2k views

### Nerve Theorem: Is the finite union of closed convex sets triangulable?

My Question: Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable$^1$? If so, why? Background: I'm trying to better understand the ...
324 views

1k views

### Conditions on $f$ such that separate continuity implies joint continuity

Consider a function $f : X\times Y \to Z$ where all spaces are compact metric spaces. Assume further that $f_x: y \mapsto f(x,y)$ and $f_y: x \mapsto f(x,y)$ are continuous. I am looking for ...
332 views

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 H_{k}(... 11 votes 0 answers 271 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 ... 10 votes 1 answer 101 views ### A topology over \Bbb N based on convergence of series. Define \tau=\{U\subseteq \Bbb N:U\in\{\Bbb N,\emptyset\}\vee\sum_{n\notin U}n^{-1}<\infty\}. In other words, a subset of \Bbb N is closed iff it is \Bbb N or the sum of the inverses of its ... 10 votes 0 answers 275 views ### Stone Duality: What are \sigma-Algebras Dual To? Stone duality, one of many dualities between certain lattices and certain topological spaces, asserts that there is a contravariant categorical equivalence between the category \text{Bool} of ... 10 votes 0 answers 397 views ### Find sequential orthographic projections, linking three different manifolds of dimension n=1,2,3 Below is an image of the family of 2D curves for reference. The image is arbitrary. Still trying to formulate a concise question. I know that the geodesics for flat Euclidean Space are straight lines. ... 10 votes 0 answers 116 views ### When is uniform space normal We know that metric spaces are normal. We also know that a uniform space is Hausdorff if the intersection of all entourages is the diagonal, in which case it is even regular. However, is there a ... 10 votes 0 answers 703 views ### The topology of GL(V) Let V be a topological vector space (not necessarily finite-dimensional) over a field K, and let GL(V) be the group of invertible linear maps V\to V under composition. There are two obvious ... 10 votes 0 answers 976 views ### Prob. 1, Sec. 27, in Munkres' TOPOLOGY, 2nd ed: How to show that the compactness of every closed interval implies the least upper bound property? Here is Prob. 1, Sec. 27, in the book Topology by James R. Munkres, 2nd edition: Prove that if X is an ordered set in which every closed interval is compact, then X has the least upper bound ... 10 votes 0 answers 242 views ### Elementary proof of compact space = exhaustible space? (This is a repost of a question I asked last year on cs.stackexchange.) The work of Martín Escardó has demonstrated close parallels between classical topology on one hand and computability on the ... 10 votes 0 answers 220 views ### Raising a partial function to the power of an ordinal Consider a set X, and let f : X \rightarrow X denote a partial function. Then for natural n, we can define f^n as iterated composition, e.g. f^2 = f \circ f. Now suppose that X is also ... 10 votes 0 answers 1k 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 ... 10 votes 0 answers 2k 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 1 answer 207 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 0 answers 107 views ### Infinite (co)-homology Lately, I've been wondering if it was possible to define singular homology also with infinite-dimensional simplices. For example we could define an infinite dimensional simplex as:$$\Delta_{\infty}:=\...
Suppose $X\subseteq\mathbb{R}^m$ s.t. for any $x\in X$ and any open $U\subseteq\mathbb{R}^m$ that contains $x$, there exists a smaller open set $V\subseteq U$ also containing $x$, so that $V\cap X$ is ...