Mathematics Stack Exchange
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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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Too much long line

There is a so-called long line in topology, which is a topological space with a base set $[0,1]\times \mathbb{R}$ with order topology given by lexicographic order: $(x_{1}, y_{1}) < (x_{2}, y_{2})$ if and only if $x_{1} < x_{2}$ or $x_{1} = x_{2}$…
Seewoo Lee
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Question about closure of the product of two sets

Let $A$ be a subset of the topological space $X$ and let $B$ be a subset of the topological space $Y$. Show that in the space $X \times Y$, $\overline{(A \times B)} = \bar{A} \times \bar{B}$. Can someone explain the proof in detail? The book I have…
user39794
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Rational numbers are not locally compact

I'm trying to show that $\mathbb Q$ is not locally compact using this definition: So I need to show that there is some point $x\in \mathbb Q$ such that no matter what neighborhood of $x$ in $\mathbb Q$ I take, no compact subset of $\mathbb Q$ can…
user557
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When is the vector space of continuous functions on a compact Hausdorff space finite dimensional?

I know that the vector space of all real valued continuous functions on a compact Hausdorff space can be infinite dimensional. When will it be finite dimensional? And how will I identify that vector space with $\mathbb{R}^n$ for some $n$?
user8018
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1 answer

Is the "evaluation map" $Y^{I}\times I\to Y$ continuous for general $Y$?

My question is explicitly the following: If $Y^{I}$ is given the compact-open topology is the map $Y^{I}\times I\to Y, (\gamma,t)\mapsto \gamma(t)$ continuous even if $Y$ is not Hausdorff? In the case that $Y$ is Hausdorff one can quickly see that…
s.harp
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Is there a continuous injection from the unit square to the unit interval?

I see that the Peano curve is a continuous surjection from the unit interval to the unit square (correct me if I'm wrong). Does it then follow that there is a continuous injection from the unit square to the unit interval? Thank you!
user46234
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3 answers

The boundary of union is the union of boundaries when the sets have disjoint closures

Assume $\bar A\cap\bar B=\emptyset$. Is $\partial (A \cup B)=\partial A\cup\partial B$, where $\partial A$ and $\bar A$ mean the boundary set and closure of set $A$? I can prove that $\partial (A \cup B)\subset \partial A\cup\partial B$ but for…
Mathematics
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Continuous functions define a topology?

The WP article on general topology has a section titled "Defining topologies via continuous functions," which says, given a set S, specifying the set of continuous functions $S \rightarrow X$ into all topological spaces X defines a topology [on…
user13618
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1 answer

Homeomorphism that maps a closed set to an open set?

In my Real Analysis class I got a bit frisky and broke out a homeomorphism in a problem to show that a set was closed (that is, I had a closed set, and I made a homeomorphism between it and the set in question to show that the set in question was…
user25326
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Can the product of two $\mathsf Y$'s be embedded in 3-space?

Let $Y$ denote the space homeomorphic to the (sans serif) letter $${\huge\mathsf Y}$$ or, equivalently, the space of three closed intervals glued together at one endpoint. Consider the space $Y\times Y$. Here is my attempt at a drawing: What I drew…
user134824
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Definition of topological space: Is Ω equal to the powerset of X?

A topological space is a set $X$ and a collection $\Omega$ of subsets of $X$ such that: $\emptyset \in \Omega$ and $X \in \Omega$ The union of any collection of $\Omega$ is in $\Omega$ The intersection of any finite collection of sets in $\Omega$…
Iordan Iordanov
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Are interior points ever limit points as well?

From my understanding of limit points and interior points there is somewhat of an overlap and that a lot of the time interior points are also limit points. For the reals, a neighborhood, $r>0$, around a point must contain only a single point of the…
allthosethings
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Normal, Non-Metrizable Spaces

We know that every metric space is normal. We know also that a normal, second countable space is metrizable. What is an example of a normal space that is not metrizable? Thanks for your help.
Daniela del Carmen
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Example of a topological space which is not first-countable

According to Munkres' Topology: Definition. A space $X$ is said to have a countable basis at $x$ if there is a countable collection $\mathscr B$ of neighborhoods of $x$ such that each neighborhood of $x$ contains at least one of the elements of…
user231343
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2 answers

How to prove the Closed map lemma

The closed map lemma says that if $f : X \to Y$ is a continuous function, $X$ is compact and $Y$ is Hausdorff, then $f$ is a closed map. How can I prove this ? Here is my attempt so far: Suppose for contradiction that $f$ is not a closed map. Then…
harlekin
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