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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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Identifying the two-hole torus with an octagon

I am aware that the 2-hole torus can be identified with the octagon with the equivalence relation as given in this picture: However, today in a topology revision lecture, the lecturer said the 2-hole torus can be represented by the octagon with…
AlexBowring
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"Area" of the topologist's sine curve

Consider the topologist's sine curve: $$ f(x) = \sin\left(1 \over x \right), x \neq 0 $$ The graph of this function resembles a space-filling curve near $x=0$. It is not a space filling curve, though, because no matter how close we get to the…
Kevin
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Definition of a Topology through neighbourhood basis?

I have a question concearning the definition of a topology through neighbourhood basis. First: Theorem 1. Let $X$ be a topological space and $x\in X$. If $\mathscr{B}(x)$ is a neighborhood basis of $x$ then: $(i)$ $\mathscr{B}(x)\neq…
PtF
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Is the unit circle $S^1$ a retract of $\mathbb{R}^2$?

In the topological sense, I understand that the unit circle $S^1$ is a retract of $\mathbb{R}^2 \backslash \{\mathbb{0}\}$ where $\mathbb{0}$ is the origin. This is because a continuous map defined by $r(x)= x/|x|$ is a retraction of the punctured…
Libertron
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A question about path-connected and arcwise-connected spaces

If $X$ is a Hausdorff topological space and it is path-connected, then it is arcwise-connected.
Summer
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$\mathbb R = X^2$ as a Cartesian product

I wonder if it is possible to consider $\mathbb R$ as a Cartesian product $X\times X$ for some set $X$. From the point of view of the dimensionality, there are spaces with a Hausdorff dimension $1/2$ (sort of Cantor sets), but I guess there are…
SBF
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What is a rigorous proof of the topological equivalence between a donut and a coffee mug?

I've seen this example given numerous times, but have never seen a real proof in a textbook.
InquilineKea
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Where do bitopological spaces naturally occur? Do they have applications?

I am interested where bitopological spaces occur in various parts of mathematics (i.e., what are natural examples of bitopological spaces stemming from various areas of mathematics, not from the studying bitopological spaces for their own sake.) I…
Martin Sleziak
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order topology is Hausdorff

Is it appropriate to ask the community to check my proof? I am rereading Munkres Topology and trying to do the HW. This is my attempt for #10 on page 101. Show that every order topology is Hausdorff. Proof: Suppose that $x_1, x_2$ are elements…
Kara
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In Euclidean space can you always find a sequence approaching a limit point along a line?

I have what seems like a very simple question: Suppose I have an open set $X \subset R^n$ and a limit point $z$ of $X$. I would like to find a sequence of points $z_1,z_2,\ldots \subset X$ approaching $z$ along a straight line ; i.e. along the line…
user167131
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Existence of embeddings does not imply the existence of a homeomorphism

Suppose there exist embeddings $f:X\to Y$ and $g:Y\to X$. Show by means of an example that $X$ and $Y$ need not be homeomorphic. I set $X=(0,1)$ and $Y=(0,\frac{1}{2})\cup (\frac{1}{2}, 1)$. I think $f:X\to Y$ defined by $f(x)=\frac{x}{2}$ is an…
Xena
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The set of lines in $\mathbb{R}^2$ is a Möbius band?

I encountered a hard-to-believe and hard-to-understand statement in this problem: Let $\mathcal{L} = \{\textrm{lines in}\ \mathbb{R}^2\}$. Consider the 2-to-1 map $f: S^1 \times \mathbb{R} \rightarrow \mathcal{L}$ given by $(\theta,x) \mapsto…
1LiterTears
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An infinite union of closed sets is a closed set?

Question: {$B_n$}$ \in \Bbb R$ is a family of closed sets. Prove that $\cup _{n=1}^\infty B_n$ is not necessarily a closed set. What I thought: Using a counterexample: If I say that each $B_i$ is a set of all numbers in range $[i,i+1]$ then I can…
jreing
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Going from a fundamental system of neighborhoods to a topology and vice versa

Given a topological space $(X,\tau)$ and a point $x\in X$ we can define a fundamental system of neighborhoods of $x$ (or perhaps a neighborhood base at $x$), say $\mathscr{N}(x)\subseteq2^X$, by every neighborhood $U$ of $x$ contains an element of…
danzibr
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What is $A\setminus U$ if $U$ is open and $A$ is closed?

Show that if $U$ is open and $A$ is closed, then $U\setminus A = \{ x\in U : x\notin A \}$ is open. What can be said about $A\setminus U$ I dont quite get why $U\setminus A = \{x\in U : x\notin A\}$ is open? If $x\in U$ and $x\notin A$ then isn't…
Paul
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