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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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How can I define a topology on the empty set?

We know that, indiscrete topology is the smallest topology. It has $2$ elements (they are the empty set and whole set). Suppose the given set is the empty set, then how can I define a topology on that set? Is it possible?
Avinash N
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Show the discrete topology is the only one larger than $\tau_l$

$(X,\le)$ is a partially ordered set, we define $U_l(x)=\{y\ |\ y\le x\}$, and $\tau_l$ is the topology generated by $\{U_l(x)\}$. We want to prove that the discrete topology is the only on that's larger than $\tau_l$. I don't understand this…
user2520938
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Prove that B is a basis for a topology

Let $X = \{0,1,2,3,\ldots\}$ (the non-negative integers), let $$B_1 = \{\{n\} : n \in X \text{ and }n > 0\}= \{\{1\}, \{2\}, \{3\},\ldots\}$$ $$B_2 = \{Z \subset X : X \setminus Z = \{1,2,\ldots n\} \text{ for some }n \in \mathbb{N} \}$$ a) Prove…
user24874
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"Abstract nonsense" proof that the fundamental group of a topological group is abelian

I seem to remember reading somewhere an "abstract nonsense" proof of the well-known fact that the fundamental group of a connected topological group is abelian. By "abstract nonsense" I mean that the proof used little more than the fact that…
Paul Siegel
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Open sets in the product topology

Let $\{X_{\alpha}\}_{\alpha\in A}$ be a family of topological spaces. The product topology on $X=\prod_{\alpha\in A}X_{\alpha}$ is the weak topology generated by the coordinate maps $\pi^{}_{\alpha}:X\to X_\alpha$. The following is an exercise about…
user9464
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Properties of closure and example

Let $A_1, A_2, A_3, \dots$ be subsets of a metric space. If $B=\bigcup_{i=1}^\infty A_i$, prove that $\overline{B}\supset \bigcup_{i=1}^\infty \overline{A_i}.$ Show, by an example, that this inclusion can be proper. Proof: We'll prove that…
RFZ
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show that the ordered square is locally connected~

show that the ordered square is locally connected but not locally path connected. what are the path components of this space? this problem is exercise munkres 25-3, and also example 24-6 and there, let $Ux$ = $f^-1(x$x$(0,1))$ and get…
solafide
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Denseness: Closed Space

I need this as lemma. Topological Space Given a topological space $\Omega$. Consider a closed space: $$\mathcal{S}\subseteq\Omega:\quad\mathcal{S}=\overline{\mathcal{S}}$$ Then for dense…
C-star-W-star
  • 16,275
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Nested Sequence of Non-Empty Compact Sets

Let $K_n$ be a nested sequence of non-empty compact sets in a Hausdorff space. Prove that if an open set $U$ contains contains their (infinite) intersection, then there exists an integer $m$ such that $U$ contains $K_n$ for all $n>m$. ... (I know…
CJQ
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If a unit circle homeomorphism commutes with an irrational rotation then it is a rotation

Let $f: S^1 \to S^1$ be an homeomorphism. I'm trying to prove that if $R\circ f = f \circ R$ where $R$ is an irrational rotation in $S^1$ then $f$ is a rotation. So, using the definition in the $[0,1]$ interval, $R(x)=x+\alpha \mod 1$ where $\alpha$…
user247322
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Prove path-lifting property using Lebesgue covering lemma

For every map $\gamma:[0,1]\to S^1$, show that there is a map $\hat\gamma:[0,1]\to\mathbb{R}$ with $\gamma(t)=P(\hat\gamma(t)),$ where $P(s)=(\cos 2\pi s,\sin 2\pi s)\in S^1$. I want to prove this proposition using Lebesgue covering lemma, which…
Toujou
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Disjoint Union Topology - understanding operations on it

I can find good explanations of how the disjoint union topology is constructed, but I am confused about how things such as complements, boundaries, limit points, etc. are to be understood in this context. For example, suppose we have two spaces, P…
ernie
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What is induced topology?

In my text, it says "Given a topological space $X$ and a subspace $S ⊂ X$, define the induced topology on $S$ to be the topology in which the open sets are of form $U ∩ S$, where $U$ is open in $X$ and $S^n$ (the n-sphere) with its induced …
Shamisen Expert
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Each discrete space is a Polish Space

I'm trying to solve exercise 6.3#7 from Sidney A. Morris' "Topology without tears": "Prove that each discrete space [...] is a Polish space." I started by proving that discrete spaces are always completely metrizable with the discrete metric. But…
jerico
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how do you prove the set of accumulation points of Q is R.

I know that the set of accumulation points for the rational numbers is the real numbers, but I'm not sure how to prove this. I need to use the definition: $x$ is an accumulation point of $S$ if, for every $y>0$, there exists a point $s$ in $S$ such…
Magen
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