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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
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Topological properties preserved by continuous maps

A continuous function does not always map open sets to open sets, but a continuous function will map compact sets to compact sets. One could make list of such preservations of topological properties by a continuous function $f$: $$ f(…
Joseph O'Rourke
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6 answers

What does "removing a point" have to do with homeomorphisms?

I am self-studying topology from Munkres. One exercise asks, in part, to show that the spaces $(0,1)$ and $(0,1]$ are not homeomorphic. An apparent solution is as follows: If you remove a point, $x$, from $(0,1)$, you get a disconnected space;…
ashman
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Closure of the union = Union of closures

I have seen that $\text{cl}(A\cup B)=\text{cl}(A)\cup \text{cl}(B)$. However I don't see why this is true. I can see the the right to left inclusion, but I can't see the inclusion from left to right. Say that I have an element $x$ contained only in…
TheGeometer
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8 answers

Example of a non-hausdorff space

For instance, the simplest example of a non-hausdorff topological space is the the pair $(X, \tau_X)$ where $\tau_X = \{ X, \varnothing \} $. But this is boring. Can someone help me find more interesting examples?
ILoveMath
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1 answer

If a topology contains all infinite subsets, then it is the discrete topology

I am solving question $8$ of exercises in section 1.1. of chapter 1 in "Topology without tears". The question reads as follows. Let $X$ be an infinite set and $\tau$ be a topology on $X$. If every infinite subset of $X$ is in $\tau$, prove that…
Adhvaitha
  • 1,991
27
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4 answers

Are continuous self-bijections of connected spaces homeomorphisms?

I hope this doesn't turn out to be a silly question. There are lots of nice examples of continuous bijections $X\to Y$ between topological spaces that are not homeomorphisms. But in the examples I know, either $X$ and $Y$ are not homeomorphic to…
Dan Ramras
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does every topology have a basis?

This might be a silly question, but i was wondering, is there any topology that cannot be generated by a basis? if not, given a topology, is there a reliable way of figuring out a basis for it? it probably matters if the set $X$ the topology is on…
Vien Nguyen
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Is there a "deep line" topological space in analogue to the "long line" $\omega_1\times[0,1)$?

I was reading about the "long line" $L=\omega_1\times[0,1)$ in the lexicographic order topology, which is locally like $\Bbb R$ except that it is "long" on one end, so there is no countable sequence that runs off to infinity unlike $\Bbb R$. My…
Mario Carneiro
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1 answer

Why do all compact connected surfaces have a triangulation?

Does anyone know a reference for a relatively elementary proof that every compact connected surface can be triangulated? By "elementary," I mean that I could present at least a sketch to undergraduates taking a first semester topology course. When…
Cheerful Parsnip
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Path-Connected implies Connected without knowing that [0,1] is connected

We all know the classical proof that a path-connected topological space $X$ is also connected. I will recall it briefly, so that we all talk about the same thing. Let $X$ be a path-connected topological space and assume for a contradiction that $X=A…
Nils Matthes
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26
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7 answers

If $A$ is a subset of $B$, then the closure of $A$ is contained in the closure of $B$.

I'm trying to prove something here which isn't necessarily hard, but I believe it to be somewhat tricky. I've looked online for the proofs, but some of them don't seem 'strong' enough for me or that convincing. For example, they use the argument…
Daavid M.
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2 answers

Topology without tears

I have started reading the book Topology Without Tears by Sidney A. Morris. I have read the first chapter and so far it reads well. However, the name of the book is a bit deceiving and makes me think it is not a book to rigorously learn topology.…
Adhvaitha
  • 1,991
25
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3 answers

Finding counterexamples: bijective continuous functions that are not homeomorphisms

Let $f: X \to Y$ be a bijective continuous function. If $X$ is compact, and $Y$ is Hausdorff, then $f:X \to Y$ is a homeomorphism. My goal is to demonstrate the necessity of both the compact and Hausdorff property of $X$ and $Y$ respectively. I…
emka
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Is it true that a subset that is closed in a closed subspace of a topological space is closed in the whole space?

I have a non homework related question from a text and require a nice clear proof/disproof please Is it true that a subset that is closed in a closed subspace of a topological space is closed in the whole space? my ideas: if $H$ is the subset of the…
Mathproof P.
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Show that a retract of a Hausdorff space is closed.

Definitions: A subspace $A \subset X$ is called a retract of $X$ if there is a map $r: X \rightarrow A$ such that $r(a) = a$ for all $a \in A$. (Such a map is called a retraction.) Problem statement: Show that a retract of a Hausdorff space is…
onimoni
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