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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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Are the following sets open subsets of $\mathbb{R}$

I need to determine whether $[2,4]$ and $\mathbb{R}-\mathbb{Q}$ are open subsets of $\Bbb R$. For $[2,4]$: I know for a subset to be open then $\forall x\in V$ $\exists (a,b)$ s.t. $x\in (a,b)\subseteq V$ I'm thinking that there's no open interval…
Gamecocks99
  • 1,023
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Given metric space $X$ with no isolated points, and dense $Y \subset X$, find $Z \subset Y$ so that $Z$ and $Y-Z$ are both dense in $X$.

The question is: Given metric space $X$ with no isolated points, and dense $Y \subset X$, find $Z \subset Y$ so that $Z$ and $Y-Z$ are both dense in $X$. Now, if a metric space $X$ with no isolated points has a countable base $B$ for the induced…
user2566092
  • 26,142
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Is the union of finitely many open sets in an omega-cover contained within some member of the cover?

Let $\mathcal{U}$ be an open cover of $\mathbb{R}$ (Standard Topology) such that $\mathbb{R} \not \in \mathcal{U}$ and for any finite set $A$ there is a $U \in \mathcal{U}$ such that $A \subseteq U$. We call such an open cover an $\omega$-cover. Can…
objectivesea
  • 823
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A question about uncountable, dense sets in R

If you have an uncountable subset of $\mathbb{R}$, which is dense in $\mathbb{R}$, is its complement countable? I'm trying to see if you can take the set of irrationals, remove a countable amount, and still have a dense subset in $\mathbb{R}$.If…
user96726
  • 49
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Natural non-trivial topology on ${\mathbb R}$ such that there are more than $2^{\mathbb N}$ open sets

The collection of open sets of ${\mathbb R}$ with standard topology only has cardinality $2^{\mathbb N}$ because we can map the collection of countable sets of intervals with rational endpoints onto the collection of open sets. Of course if we…
user2566092
  • 26,142
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group of homeomorphisms of X

Let $Homeo(X)$ denote the group of homeomorphisms of a compact Hausdorff space $X$ onto itself. If $Homeo(X)$ is isomorphic to $Homeo(Y)$ (as topological groups with compact-open topology), can we conclude that $X$ is homeomorphic to $Y$ ?
nsoum
  • 894
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Base of clopen and $T_{0}$ implies $X$ is Tychonoff

Let $X$ be a topological space and assume $X$ has a base $\mathcal{B}$ of clopen sets. Show $X$ is completely regular and a $T_{0}$ space. My try: First it is not hard to show that if $B \subset X$ then $\chi_{B}$, the characteristic function of $B$…
user10
  • 5,688
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A complete proof that a triangle (or arbitrary polygon) is a cell

A cell is any subset of the plane homeomorphic to a disk. Could someone provide a complete proof that a triangle is homeomorphic to a disk? I have two ideas but I can't seem make them fully rigorous. One would be via this explicit mapping $f$: Map…
Harry Stern
  • 861
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Characterization of totally bounded sets in Metric spaces

Searching material to know more about totally bounded sets, I found this property Let $(M,d)$ be a metric space. So $A\subset M$ is totally bounded if and only if for every $\varepsilon >0$ exists a compact $K$ such that $d(x,K)<\varepsilon$ for…
Juan Pablo
  • 626
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Base and empty set of a topology

Given space $X = \{a, b, c\}$, $\beta$ is a basis for a topology $\tau$ on X. $\tau = \{ \varnothing, X, \{a\}, \{b\}, \{a,b\}\}$, $\beta = \{\{a\}, \{b\}, X\}$. $\beta$ can't union its elements to get empty set $\varnothing$ contained in…
Matt Elson
  • 255
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A topology generated by neighborhoods

If $X$ is a topological space, we say that $U$ is a neighborhood of $x$ if there exists an open set $V$ such that $x\in V\subseteq U$. Let $X\neq\emptyset$ and suppose that for every $x\in X$ there exists $\mathcal{V}(x)$ a family of subsets of $X$…
Talexius
  • 2,015
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If you lived in a 4-torus, what would the doughnut hole look like from the inside?

I'm not just curious; it refers to general relativity. Specifically, would the hole in the torus' center look to us like a sphere, one you cannot enter because you always slip across the side and go around it instead of through?
Duce ex Machina
  • 250
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Understanding the Solid torus

I have asked this same question and it was closed in less than two minutes, I really don't understand why is it very stupid? (sorry for being very stupid). I've come across the following exercise: Prove that the solid torus denoted $\tau$ which is…
Asma
  • 371
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What's wrong with the following false-proof that a subspace of a normal space is normal?

The fact that a subspace of a normal topological space is normal is known to be wrong. I'm trying to find the wrong step in the following proof: Let $X$ be a normal topological space and $Y$ be a subspace. Let $C_1, C_2$ be closed sets in $Y$. …
Omer
  • 2,490
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1 answer

Surjective local homeomorphisms from X to Y and from Y to X; are X and Y homeomorphic?

Problem Let $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ be topological spaces, and $f : X \to Y$ and $g : Y \to X$ be surjective local homeomorphisms. Is $\mathcal{T}_X$ homeomorphic to $\mathcal{T}_Y$? Background For background, see here. Since…
kaba
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