Mathematics Stack Exchange
Mathematics Stack Exchange

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.

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; precompactness; separation axioms;countability axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; proximity; etc. Please use the more specific tags, , , , , whenever appropriate.

57719 questions
4
votes
3 answers

Prove the int(int(S)) = int(S)

I had a question on what exactly I need to show in proving that the interior of the interior of a set is equal to the interior of a set. I'm given a set $A\subset X$, where $X$ is a topological space, and I want to show $int(A) = int(int(A))$, where…
rohan
  • 41
4
votes
2 answers

Why topology is called Rubbersheet Geometry?

Usually topology classes starts with comparing doughnut and tea cup. But after introductory class teacher will move to the definition of topology as a collection of subsets of a set having certain properties.. At what point does this meet with our…
Madhu
  • 1,695
4
votes
1 answer

If $X$ is a compact metric space, $Homeo(X)$ is second countable

I would like an easy way to show that if $X$ is a compact metric space, then the group $Homeo(X)$ of homeomorphisms of $X$ with the compact-open topology (or the topology of uniform convergence, they are the same in this case) is second countable. I…
Nicolas
  • 355
4
votes
1 answer

Does homeomorphism preserve second countabity?

This seems obviously true and i proved it, but i couldn't find this in googls, so i'm asking this to make sure. Let $X,Y$ be topological spaces. Let $H:X\rightarrow Y$ be a homeomorphism. Is $Y$ second-countable if $X$ is second-countable?
John. p
  • 1,663
4
votes
3 answers

Why is $[0,1]^{[0,1]}$ not first countable?

In a ps file on topological properties, the set $[0,1]^{[0,1]}$ is given as an example of a product topology that is not first countable. Is there a proof of why?
Roofuss
  • 41
4
votes
1 answer

Proving that $\Bbb R^{n} \setminus S$ is connected if $S$ is countable

Let $n>2$. Let $S$ be a set that is at most countable. Prove, that $\Bbb R^{n} \setminus S$ is a connected set. Let's start $n=2$. How to show it in a formal way?
user116011
  • 363
4
votes
1 answer

Object with holes.

If you take a sphere and drill a hole through it, the shape can be continuously deformed to a torus of genus = 1. A sphere with two separate holes is homeomorphic to a torus with genus 2. I am wondering about a similar kind of shape where you…
user18764
  • 216
4
votes
1 answer

Verifying the subspace topology with a topology defined in terms of neighborhoods

In M.A. Armstrong's Basic Topology, he introduces the concept of a topological space by first defining it in terms of neighborhoods (rather than open sets). Here is his formulation (p. 13): We ask for a set $X$ and for each point $x$ in $X$ a…
dwar
  • 253
4
votes
3 answers

Continuity via Closure Operator

I'm having a rather simple question: Lets say a function preserves neighborhoods iff: $N\in\mathcal{N}_x \Rightarrow f^{-1}(N)\in\mathcal{M}_x$ and a function preserves closeness iff: $x\parallel A \Rightarrow f(x)\parallel f(A)$ I want to show…
C-star-W-star
  • 16,275
4
votes
1 answer

Middle of mobius band is a circle

We define the Möbius band $M$ as the space $I \times I$ when identifying $(0,t)\sim (1,1-t)$ through $\rho: I \times I \to M$. We wish to prove $S=\rho(I \times \{ \frac{1}{2}\})$ is a circle. The solution was to define a function $f:I \times…
Emolga
  • 3,527
4
votes
1 answer

Neighborhood of a point in Topology

Let $x \in X$ and define a topological space $(X, \tau)$ and let singleton set {$x$} $\in \tau$. Then by definition of neighborhood of a point in topology, {${x}$} will be a neighborhood of point $x$. My question is If set {$x$} does not contain…
Andrew Miller
  • 812
4
votes
0 answers

(Dis)continuity of stepfunction in topology

I'm trying to learn a little about topology, and I don't quite understand continuity yet. I use this definition of a continuous map f: f is continuous if the inverse image of every open set is open. I use the sets $\mathbb{R}$ and $A =…
user119879
  • 151
4
votes
1 answer

Give an example of a local homeomorphism $R^2 → S^2$ which is surjective.

So this is a qual problem and I can't think of an easy obvious map that will do this. Any help will give you good karma I'm sure. Give an example of a local homeomorphism $R^2 → S^2$ which is surjective.
kingkongdonutguy
  • 1,713
4
votes
2 answers

Basis for a topology with a countable number of sets

I am working on the problems from the textbook "Topology without tears". I am stuck with problem number $4$ in Exercise $2.2$. Could anyone suggest some hints on how to proceed? The question goes as follows. A topological space $(X,\tau)$ is said to…
Adhvaitha
  • 1,991
4
votes
2 answers

Example of a non locally-compact space with a dense locally compact subspace.

Let $X$ be a topological space such that $A \subset X$ is a dense subspace which is locally compact and $B \subset X$ is a dense subspace which is not locally compact (at all of its points). Is it possible to find such $X$? if it is, an example?
user116457
1 2 3
…
99 100