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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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Verifying a Closure Operation

I'm just starting to learn some basic topology, and I've mostly encountered definitions. For instance, I read that for a topological space $X$, for any $E\subseteq X$, define its closure $\overline{E}$ as the set of points $p\in X$ such that each…
yunone
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Show that two discrete spaces are homeomorphic iff they have the same cardinality

I have the following question: Show that two discrete spaces are homeomorphic iff they have the same cardinality: I have tried the following: Let $f: (X, \mathcal{T}_{discrete}) \to (Y, \mathcal{T}_{discrete})$ $(\Rightarrow )$ As we have that the…
LFRC
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Why does having fewer open sets make more sets compact?

In functional analysis and in PDE theory, we are interested in proving existence results. Such results are generally obtained on some compact space for some given topology. And this is the reason why in functional analysis, we are always…
Benjamin
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How to prove that : $Z$ is closed subset iff $X$ can be covered by open subsets $U$ such that $Z\cap U$ is closed in $U$ for each $U$

Let $X$ be a topological space, $Z$ is closed subset of $X$ if and only if $X$ can be covered by open subsets $U$ such that $Z\cap U$ is closed in $U$ for each $U.$ Can someone help me to prove this or give me a reference where I can find it,…
user122327
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The uniform metric on $\mathbb{R}^\omega$

I'm working on a question from Topology by Munkres on p. 127 (Exercise #6). Here it is: Let $\bar{\rho}$ be the uniform metric on $\mathbb{R}^\omega$. Given $\mathbb{x}= (x_1, x_2, ...) \in \mathbb{R}^\omega$ and given $0 < \epsilon < 1$, let…
Libertron
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Quotient of zero-dimensional hausdorff space

I've read (in joy of cats) that every topological space is a regular quotient of a zero-dimensional hausdorff space. So far, I could not find a proof. Do you know one, or a reference?
Martin Brandenburg
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Range of curve on a compact interval is nowhere dense

I glanced through this question on why $\mathbf{R}^2$ is not of the first category. I understand how this would follow if the image of a curve on a compact/finite interval in $\mathbf{R}$ is nowhere dense in $\mathbf{R}^2$. I didn't understand any…
Pierre R.
  • 157
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What is meant with unique smallest/largest topology?

I'm doing this exercise: Let $\{T_\alpha\}$ be a family of topologies on $X$. Show that there is a unique smallest topology on $X$ containing all the collections $T_\alpha$, and a unique largest topology contained in all $T_\alpha$. I have…
Kasper
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A topology generated by sets

Let $F$ be a family of sets. It is possible to define a topology $T$ generated by $F$ by letting it the intersection of all topologies containing $F$. If $F$ is known it is also possible to construct $T$ as follows: (1) add $F$, $\varnothing$ and…
blue
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Is there a continuous injective map from $\mathbb{R}$ that has compact image?

Suppose I have a function $f: \mathbb{R} \to X$, where $X$ is some non-compact metric space. Is it possible that $f$ is injective yet has compact image? If the answer is yes, what characterizes those cases? What additional properties would make the…
Ofir
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Prove X has the discrete topology, given every point is open?

I was wondering what would be sufficient to show that $X$ has a discrete topology. I know the following: $X$ is a topological space where each point $x$ is open ($\{x\}$ is open for each $x\in X$), and I want to show that $X $ has the discrete…
graham
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Can you conjecture what functions $f: \mathbb{R} \to \mathbb{R}$ is continuous when considered as maps from $\mathbb{R}$ to $\mathbb{R_\mathcal{l}}$?

The complete question is from Mukres's Topology. (a) Suppose that $f: \mathbb{R} \to \mathbb{R}$ is "continuous from the right" that is $$\lim_{x \to a^{+}} f(x) = f(a),$$ for each $a \in \mathbb{R}$. Show that $f$ is continuous when considered as…
Jichao
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How to prove that this set is closed?

Suppose $Y$ be an ordered set in the order topology. Let $f, g: X \to Y$ be continuous. How to show that the set $\{x| f(x) \leq g(x)\}$ is closed? This is a excercise from munkres. Maybe trying to show that the set $f(\{x| f(x) > g(x)\})$ is open…
Jichao
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Topology exercise. Quotient Space

I found this exercise I'm not able to solve. EXERCISE : Let's call $X$ the topological space obtained quotienting $\mathbb{R}^{n}$ by the equivalence relation $\sim$ : $x\sim y \Leftrightarrow x=y$ or $||x||=||y||$ or $||x||\cdot ||y||=1$. Is $X$ an…
Lorenzo Rossi
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Does there exist a compact Hausdorff topology on the natural numbers?

I was wondering whether there exists a compact Hausdorff topology on $\mathbb N$. The only result I was able to find in this context was that, if a set has a topology that is compact, Hausdorff and has no isolated points then the set is uncountable.…
skf23852
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