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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
155
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Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$

It is very elementary to show that $\mathbb{R}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>1$: subtract a point and use the fact that connectedness is a homeomorphism invariant. Along similar lines, you can show that $\mathbb{R^2}$ isn't…
user7530
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88
votes
1 answer

Does $X\times S^1\cong Y\times S^1$ imply that $X\times\mathbb R\cong Y\times\mathbb R$?

This question came up in a recent video series of lectures by Mike Freedman available through Max Planck Institut's website. He proves the "difficult" converse direction, that $X\times \mathbb R\cong Y\times \mathbb R$ implies $X\times S^1\cong…
Cheerful Parsnip
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73
votes
12 answers

Why is empty set an open set?

I thought about it for a long time, but I can't come up some good ideas. I think that empty set has no elements,how to use the definition of an open set to prove the proposition. The definition of an open set: a set S in n-dimensional space is…
python3
  • 3,494
72
votes
4 answers

Difference between basis and subbasis in a topology?

I was reading Topology from Munkres and got confused by the definition of a subbasis. What is/are the difference between basis and subbasis in a topology?
Grobber
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64
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15 answers

Why can't you flatten a sphere?

It's a well-known fact that you can't flatten a sphere without tearing or deforming it. How can I explain why this is so to a 10 year old? As soon as an explanation starts using terms like "Gaussian curvature", it's going too far for the audience…
Joe
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62
votes
5 answers

When is the closure of an intersection equal to the intersection of closures?

We know $\overline{\bigcap A_{\alpha}}\subseteq\bigcap\overline{A}_{\alpha} $, but when is the reverse inclusion true? Can you give some properties of the underlying space that would guarantee this?
Forever Mozart
  • 9,294
59
votes
3 answers

Path connectedness and locally path connected

The Section on Covering Maps in John Lee's book "Introduction to Smooth Manifolds" starts like this: Suppose $\tilde{X}$ and $X$ are topological spaces. A map $\pi : \tilde{X} \to X$ is called a covering map if $\tilde{X}$ is path-connected and…
harlekin
  • 8,740
56
votes
3 answers

Trying to define $\mathbb{R}^{0.5}$ topologically

A few days ago, I was trying to generalize the defintion of Euclidean spaces by trying to define $\mathbb{R}^{0.5}$. Question: Is there a metric space $A$ such that $A\times A$ is homeomorphic to $\mathbb{R}$? I am interested also in seeing…
Amr
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52
votes
6 answers

What is a topological space good for?

I know there are already some questions similar to this, which all give an answer that a topological space creates some structure on a set which is an abstraction of distance and makes it possible to define other concepts like connectedness,…
holistic
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50
votes
3 answers

What is the Topology of point-wise convergence?

What is the Topology of point-wise convergence? It has been stated in lectures but I am unfamiliar with it.
user58514
48
votes
2 answers

Existence of non-constant continuous functions

Under what circumstances is there at least one non-constant continuous function from a topological space $X$ to a topological space $Y$? Assume that $X$ and $Y$ each have at least two points. If $X$ is disconnected, separated by $A$ and $B$, then…
dfeuer
  • 9,069
46
votes
6 answers

Are the rationals a closed or open set in $\mathbb{R}$?

I have a feeling they are neither closed nor open as the $\mathbb{R} \setminus \mathbb{Q}$ cannot be open or closed either...
user26069
42
votes
1 answer

Are contractible open sets in $\mathbb{R}^n$ homeomorphic to $\mathbb R^n$?

Is it true that every contractible open subset of $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^n$?
J-Holomorph
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42
votes
4 answers

Why is the Hilbert Cube homogeneous?

The Hilbert Cube $H$ is defined to be $[0,1]^{\mathbb{N}}$, i.e., a countable product of unit intervals, topologized with the product topology. Now, I've read that the Hilbert Cube is homogeneous. That is, given two points $p, q\in H$, there is a…
Jason DeVito - on hiatus
  • 50,257
39
votes
7 answers

Does an uncountable discrete subspace of the reals exist?

Does an uncountable and discrete subspace of the reals exist?
Daniel Dreiberg
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