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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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Do simply-connected open sets admit simply connected compact exhaustions?

Suppose $E$ is a simply-connected open subset of $\mathbb R^n$. Must there be a sequence of compact subsets $K_n$ such that $E = \bigcup_{n=1}^\infty K_n$, $K_n \subseteq K_{n+1}$ for all $n$, and each $K_n$ is simply connected? This is trivially…
E.Y.Jaffe
  • 173
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Any finite set is compact; what exactly is a finite set?

Any compact set is finite. Assume the sets are in $\mathbb{R}$ Since $A = [0, 1]$ is compact, it is also finite. As for $B = (0, 1)$, it is not compact, so it is infinite. However, how is it infinite? There are infinitely many points between 0 and…
Adrian
  • 1,976
7
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Boundary of Boundary of a set?

I have reading about the closure interior and boundary operators in a topological space. I have been thinking about the following: If $ b $ denotes boundary operator and $c$ , $i$ and $k$ denote closure and interior and complement respectively. Let…
User23
  • 291
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Is every path-connected open subset of $\mathbb{R}^2$ homeomorphic to $\mathbb{R}^2$?

It is a standard result that the open ball $$B^2=\{(x,y)\in\mathbb{R}^2:x^2+y^2<1\}$$ is homeomorphic to $\mathbb{R}^2$ itself. Also, distorting $B^2$ by any continuous bijective transformation will also give a path-connected open set homeomorphic…
Chern
  • 73
7
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2 answers

'Equivalent' Exhaustion by compact sets

Given an open set $U \subset \mathbb R ^n $, there exists an exhaustion by compact sets, i.e. a sequence of compact sets $K_i$, s.t. $\cup _{i=0}^{\infty} K_i = U$ and $\forall i \in \mathbb N : K_i \subset K_{i+1} ^{\circ}$ We can imagine that…
shuhalo
  • 7,485
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Is a one point space Hausdorff?

This may be a silly question but I'll ask anyway: Is a topological space consisting of a single point Hausdorff?
iwriteonbananas
  • 6,102
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Are neighbourhood and open neighbourhood always exchangeable in statements?

From Wikipedia If $X$ is a topological space and $p$ is a point in $X$, a neighbourhood of $p$ is a subset $V$ of $X$, which includes an open set $U$ containing $p$, Note that the neighbourhood $V$ need not be an open set itself. If $V$ is…
Tim
  • 47,382
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1 answer

Why is it hard to prove Jordan Curve Theorem in the case of Koch snowflake

Many books and papers mentioned that it is easier to prove Jordan Curve Theorem in the case of polygon and hard in the case of badly behaving curves. One example that most give is Koch snowflake. My question is specifically about Koch snowflake. Why…
LedgeDash
  • 73
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Infinite removal of isolated points from a subset of $\mathbb{R}$.

Assume we have subset of $\mathbb{R}$. Then, at every step, we remove all isolated points from what is remained from initial subset. We stop when there is nothing to remove - so current set is either empty or there is no isolated points. Is there a…
Hedgehog
  • 1,722
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2 answers

Can a dense set contain isolated points?

I was interested in this question, can a dense set contain an isolated point, because I was reading into the lexicographic order topology on the unit square. I read in here that: $S$ is not separable, since the set of all points of the form…
Eric_
  • 935
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2 answers

Covering $\mathbb{R}^2$ with uncountably many disjoint non-degenerate line segments

Is it possible to cover $\mathbb{R}^2$ with uncountably many disjoint non-degenerate line segments? If a formal definition is necessary, let's define a line segment as a set $\{(x, mx+c): x \in [a, b]\}$ for some fixed constants $m, c, a, b \in…
donburi
  • 836
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2 answers

Meaning of a discrete topological sub-space?

Given a topological space $X$ and a set $U\subseteq X$, what is the meaning of $U$ being a discrete sub-space of $X$? I do know what a discrete space is, so as far as I understand it, the meaning is that each $A\subseteq U$ is open in the relative…
Eric_
  • 935
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votes
1 answer

A function with zero gradient is locally constant

A mapping $f:X\to Y$ is defined to be locally constant if $\forall x\in X$, there exists a neighbourhood $V(x)$ containing $x$ such that $a\in V(x)\implies f(a)=x_0$ for some constant $x_0$. In other words, every point in that neighbourhood maps to…
user67803
6
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1 answer

Unique Limits in T1 Spaces

It's intuitive to me that limits in T2 (Hausdorff) spaces are unique: $x_n \rightarrow l$ if you can find an $N$ such that for $n > N$, $x_n \in O$ where $O$ is any open neighborhood of $l$ and since distinct points in T2 spaces always have disjoint…
tac-eibmoz
  • 161
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2 answers

Is a direct proof of this possible

Consider the following statement $x_n \to x$ if and only if every subsequence of $x_n$ has a subsequence that converges to $x$. $\implies$ is clear. A proof of the other direction is given here. It is a proof by contrapositive. My question is:…
user167889
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