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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.

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Is $[0,1]$ a countable disjoint union of closed sets?

Can you express $[0,1]$ as a countable disjoint union of closed sets, other than the trivial way of doing this?
Kevin Ventullo
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Does this property characterize a space as Hausdorff?

As a result of this question, I've been thinking about the following condition on a topological space $Y$: For every topological space $X$, $E\subseteq X$, and continuous maps $f,g\colon X\to Y$, if $E$ is dense in $X$, and $f$ and $g$ agree on $E$…
Arturo Magidin
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102
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Continuous bijection from $(0,1)$ to $[0,1]$

Does there exist a continuous bijection from $(0,1)$ to $[0,1]$? Of course the map should not be a proper map.
Alex
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101
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5 answers

Defining a manifold without reference to the reals

The standard definition I've seen for a manifold is basically that it's something that's locally the same as $\mathbb{R}^n$, without the metric structure normally associated with $\mathbb{R}^n$. Aesthetically, this seems kind of ugly to me. The real…
user13618
100
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5 answers

What concept does an open set axiomatise?

In the context of metric (and in general first-countable) topologies, it's reasonably clear what a closed set is: a set $F$ is closed if and only if every convergent sequence of points in $F$ converges to a point also in $F$. This naturally…
Zhen Lin
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When is the closure of an open ball equal to the closed ball?

It is not necessarily true that the closure of an open ball $B_{r}(x)$ is equal to the closed ball of the same radius $r$ centered at the same point $x$. For a quick example, take $X$ to be any set and define a…
Alex Lapanowski
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96
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Explain "homotopy" to me

I have been struggling with general topology and now, algebraic topology is simply murder. Some people seem to get on alright, but I am not one of them unfortunately. Please, the answer I need is ideally something very elaborate and extensive,…
John Trail
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A and B disjoint, A compact, and B closed implies there is positive distance between both sets

Claim: Let $X$ be a metric space. If $A,B\subset X$ are disjoint, $A$ is compact, and $B$ is closed, then there is $\delta>0$ so that $ |\alpha-\beta|\geq\delta\;\;\;\forall\alpha\in A,\beta\in B$. Proof. Assume the contrary. Let $\alpha_n\in…
Benji
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Projection map being a closed map

Let $\pi: X \times Y \to X$ be the projection map where $Y$ is compact. Prove that $\pi$ is a closed map. First I would like to see a proof of this claim. I want to know that here why compactness is necessary or do we have any other weaker…
M.Subramani
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$X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed

Let $X$ be a topological space. The diagonal of $X \times X$ is the subset $$D = \{(x,x)\in X\times X\mid x \in X\}.$$ Show that $X$ is Hausdorff if and only if $D$ is closed in $X \times X$. First, I tried to show that $X \times X \setminus D$…
Br09
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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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Why is compactness in logic called compactness?

In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set. Most logic texts either don't explain the terminology, or allude to the topological property of compactness. I…
vanden
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Continuous mapping on a compact metric space is uniformly continuous

I am struggling with this question: Prove or give a counterexample: If $f : X \to Y$ is a continuous mapping from a compact metric space $X$, then $f$ is uniformly continuous on $X$. Thanks for your help in advance.
the code
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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. Can someone clear the sentence to me?
Sepideh Bakhoda
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Why is the Möbius strip not orientable?

I am trying to understand the notion of an orientable manifold. Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that $\textrm{det}(J(\phi_{\alpha} \circ…
Richard G
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