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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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Prove a square is homeomorphic to a circle

$s:=\{|x|\le 1,|y|\le 1\} $ $c:=\{{x}^{2}+{y}^{2}\le1\}$ Prove $\overset{\circ}{s} \cong \overset{\circ}{c}$ ok... not to sure what to do. I think $\overset{\circ}{s} \to\overset{\circ}{c}$ is something…
andwil
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22
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6 answers

Singleton sets are closed in Hausdorff space

How can I see that singleton sets are closed in Hausdorff space? That is, why is $X\setminus \{x\}$ open?
HBHSU
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22
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4 answers

Is $[0,\infty)$ a closed subset?

A closed set is one where it contains all it's limit points, even though the end is 'open' as in the traditional sense any sequence tending to infinity will never leave the subset; therefore it's closed. Is my logic sound? Thanks!
user26069
21
votes
6 answers

Showing that an open interval is an open set

Context: I'm trying to algebraically prove that an open interval is an open set. If I sketch it, as suggested by @rschwieb in this answer, then it seems quite obvious that this is indeed true. But I would like to be able to show it algebraically and…
Hunter
  • 1,309
21
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3 answers

Closure of interior of closed set

If $D$ is a closed set, what is the relation in general between the set $D$ and the closure of $\operatorname{Int}D$? We know that $\operatorname{Int}D\subseteq D$, so $\overline{\operatorname{Int}D}\subseteq \overline{D}$, but since $D$ is…
PJ Miller
  • 8,193
21
votes
3 answers

Proof that convex open sets in $\mathbb{R}^n$ are homeomorphic?

This is an exercise from Kelley's book. Could someone help to show me a proof? It seems very natural, and it is easy to prove by utilizing the arctan function in $\mathbb{R}^1$. Thanks a lot.
XxXxX
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20
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1 answer

How do Slinkies become tangled?

The following image describes the problem better than I can: As you know, sometimes Slinkies can twist such that the direction of the coil can be reversed. However, though reversed, the coil still maintains its radius, number of turns, and shape.…
user14069
  • 1,305
20
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1 answer

About the interior of the union of two sets

Let $X$ be a topological space and let $A,B\subseteq X.$ We know that in general, we only have $$\operatorname{int}(A)\cup\operatorname{int}(B)\subseteq\operatorname{int}(A\cup B). $$ My question is: When do we say that equality holds? I came with…
Juniven Acapulco
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20
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1 answer

Must every subset of $\mathbb R$ contain $2$ homeomorphic distinct open sets?

Take $X\subseteq\mathbb R$ containing at least $2$ points, I want to prove or construct a counterexample to the statement "$X$ contains $2$ distinct open sets which are homeomorphic" (where $X$ is equipped with the induced topology and $\mathbb R$…
Alessandro Codenotti
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20
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4 answers

How to prove closure of $\mathbb{Q}$ is $\mathbb{R}$

How to prove closure of $\mathbb{Q}$ is $\mathbb{R}$? Can it be proved easily?
Vinod
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20
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3 answers

What Topology does a Straight Line in the Plane Inherit as a Subspace of $\mathbb{R_l} \times \mathbb{R}$ and of $\mathbb{R_l} \times \mathbb{R_l}$

Given a straight line in the plane, what topology does this straight line inherit as a subspace of $\mathbb{R_l} \times \mathbb{R}$ and as a subspace of $\mathbb{R_l} \times \mathbb{R_l}$, where $\mathbb{R_l}$ is the lower limit topology? So trying…
Set
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19
votes
2 answers

Prove $\epsilon$-$\delta$ definition of continuity implies the open set definition for real function

I need to prove that the $\epsilon$-$\delta$ definition of continuity implies the open set definition continuity for a real function. Here's my attempt. For any basis $V: (a, b)$ in the range, for each $f(x) \in V$, let $\epsilon = \min(f(x) - a, b…
Jichao
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19
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2 answers

Is a topology recoverable from its set of neighborhoods?

Let $(X, \tau)$ be a topological space and $\tau$ be the set of opens. Let's call a pair of the form $(x, W)$ where $x$ is a point in $X$ and $W$ is a (not necessarily open) neighborhood of $x$ a neighborhood pair. Let's call the collection of all…
Greg Nisbet
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19
votes
2 answers

If totally disconnectedness does not imply the discrete topology, then what is wrong with my argument?

Assume that $X$ is a totally disconnected space. Then every two-point set is disconnected, which implies that every singleton is open in the topology of $X$ (because the one-point subsets of two-point sets form a separation). Isn't the collection of…
Xena
  • 3,853
19
votes
2 answers

The closure of an irreducible subset of an irreducible space is irreducible.

Start with an irreducible space $X$. Take a subset $Y$ that is irreducible. Show that the closure of $Y$ is still irreducible. I imagine we are supposed to start with saying, assume we have a decomposition for $\bar Y = S\cup T$ and then somehow…
Steven-Owen
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