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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.

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An irrational flow on a torus is dense

I was surprised I couldn't find the proof of this here. The problem is to prove the image of $\{(r,r\sqrt 2)\mid r\in\mathbb R\}$ is dense in the torus where we think of the torus as $I\times I$ with opposite edges identified in the usual way,…
Gregory Grant
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Theorem 20.4 in Munkres' TOPOLOGY, 2nd edition: How are these three topologies different on an infinite Cartesian product of $\mathbb{R}$ with itself?

The standard topology on $\mathbb{R}$, the set of real numbers, has as a basis all open intervals $(a,b)$, where $a$ and $b$ are real numbers such that $a < b$. Let $J$ be an arbitrary (finite, countable, or uncountable) index set, and let…
Saaqib Mahmood
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Homeomorphisms in $\mathbb{R}^2$.

I'm getting confused about homeomorphisms. I believe that $[0,1]\times [0,1)$ and $[0,1)\times [0,1)$ are homeomorphic but $[0,1]\times (0,1)$ and $[0,1)\times (0,1)$ are not. Please can you try and explain why this is the case?
Daniel Ward
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Is the following set connected given that the union and intersection is connected

Suppose $U_1, U_2$ are open sets in a space $X$. Suppose $U_1 \cap U_2$ and $U_1 \cup U_2$ are connected. Can we conclude that $U_1$ must be connected?? I am trying to find a counterexample, but I failed. PErhaps it is true? Can someone help me find…
user139708
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Can any space be seen as the boundary of some other space?

For example two points can be seen as the boundary of a line segment in $\mathbb{R}$. The plane $\mathbb{R}^2$ can be seen as the boundary of some set in $\mathbb{R}^3$ and so on. Can any set be seen as the boundary of some other set?
Cristian Em.
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Is $T=\{U\cup (V \cap \mathbb{Q}), U,V$ open in $\mathbb{R} \}$ a topology on $\mathbb{R}$?

Let $T$ be the set of all subsets of $\mathbb{R}$ of the form $U\cup (V \cap \mathbb{Q})$ where $U$ and $V$ are open in the usual topology on $\mathbb{R}$. Is $T$ a topology? Is it Hausdorff? I attempted going through the 3 conditions that a…
hotstuff69
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Topological Spaces and Topological Equivalence

My professor gave us the following question while covering Topological Spaces Find a list of Topologies $\mathcal{T}_1, \mathcal{T}_2,..., \mathcal{T}_n$ on $X=\{1,2,3\}$ such that for every topology $\mathcal{U}$ on $X$, $(X,\mathcal{U}) \equiv…
Kevin_H
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Topology, Basis of a given topology.

We defined basis for a topology, and there is something that I do not understand. Here is how we defined the basis. Given a topological space $\left(X,\mathscr T\right)$ we defined basis for the topology to be the set $\mathscr B$ , consisting of…
Charles Carmichael
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Question about a closed mapping.

We just learned about quotient mappings and various properties of the quotient topology. I'm curious about metrizability under these mappings. Namely, if $f: X \rightarrow Y$ is a closed continuous surjection and $X$ is metrizable, does it follow…
josh
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Alexandrov Spaces or Topology on a finite point set - is there a distinction between open and closed sets?

A topology can be defined as a family of sets that is closed under finite intersections and unions. We then call these open sets. But, if the point-set is finite, why should we call these open sets, and not closed sets? Consider: The union of any…
Jonathan Hebert
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Closed image of locally compact space

If $X$ is locally compact and $f : X \rightarrow Y$ is continuous closed and surjective, must $Y$ be locally compact? This seems like it should be relatively simply to answer, but I am unable to find either a proof or a counterexample. Any ideas?
user15464
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How to show 2 bases generate the same topology?

I don't see anything in my topology references to show that 2 bases generate the same topology. Is there a criteria?
Guest_topology
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Definition of uniform equicontinuity of a family of mappings between uniform spaces

From Wiki A set $A$ of continuous functions between two uniform spaces $X$ and $Y$ is uniformly equicontinuous if for every element $W$ of the uniformity on $Y$, the set $\{ (u,v) ∈ X × X: \forall f ∈ A. (f(u),f(v)) ∈ W \}$ is a member of the…
Tim
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Are the sets in a basis open in the topology the basis generates?

A collection $\mathcal{B}$ of subsets of a set $X$ is called a basis if: For each $x\in X$, $\exists B\in\mathcal{B}$ with $x\in B$. If $x\in B_1\cap B_2$ for $B_1,B_2\in\mathcal{B}$, then $\exists B_3\in\mathcal{B}$ with $x\in B_3\subset B_1\cap…
Alec Teal
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Why can't suspace topologies be the empty set?

Let $(X,\mathcal{J})$ be a topological space and let $Y\subset X$, Define the collection $\mathcal{J}'$ of subsets of $Y$ as $\mathcal{O}'\subset Y$ of the form $\mathcal{O}'=\mathcal{O}\cap Y$ where $\mathcal{O}\in\mathcal{J}$ Then…
Alec Teal
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