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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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Continuous extension of Euclidean spaces?

I am wondering if it is possible to "continuously" increase the dimension of Euclidean spaces — in other words, would it be possible to define Euclidean spaces of non-integer dimensions with nice topological properties? I have thought about the way…
user402005
  • 51
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Closure of $\mathbb{Q}\times\mathbb{Q}$ in British Rail metric

I'm wondering what is the closure of $\mathbb{Q}\times\mathbb{Q}$ in $(\mathbb{R}^{2},d)$ where $d$ is British Rail metric: $$ d(x,y) = \left\{ \begin{array}{lr} ||x-y|| & \text{if} \; \; x,y,0 \; \; \text{are collinear,}\\ || x || + ||y||& \;\;\;\;…
zinsek
  • 103
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Prove that $\mathcal T$ is a topology

Let a function $h:\mathcal P(X)\to\mathcal P(X)$ be defined by $h(\emptyset)=\emptyset$ $h(A\cup B)=h(A)\cup h(B),\;\forall A,B\in\mathcal P(X)$ $h(A)\supseteq A,\;\forall A\in\mathcal P(X)$ $h\circ h=h$ Now setup $\mathcal…
Masacroso
  • 30,417
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Closure of $\{ (x_n) | x_n = 0 $ for almost all $n \}$

What is the closure of $\{ (x_n) |  x_n = 0 $ for almost all $n \}$ in $\mathbb{R}^\mathbb{N}$ with the product topology? I'm not sure how to think about it. A point $x$ is in the closure of a set $A$ if for every neighbourhood $N$ of $x$, $N \cap…
Rudy the Reindeer
  • 46,359
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What does $\text{cl}(\text{im}(e_\mathbb{N}))\subseteq 2^{2^{\mathbb{N}}}$ look like?

This is a follow-up question to that question. Let $2 = \{0,1\}$ be endowed with the discrete topology. My hunch was the following: Let $e_\mathbb{N}: \mathbb{N} \to 2^{2^{\mathbb{N}}}$ be the "evaluation map", that is, it is given by…
Dominic van der Zypen
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Product of topological spaces and Kuratowski closure axioms

(Both finite and infinite) product of topological spaces are often expressed through open sets. Can it be expressed instead in terms of Kuratowski closure axioms (directly in terms of Kuratowski axioms, not through the isomorphism between the…
porton
  • 5,053
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A characterization of proper maps

Let $M,N$ topological spaces, $F:M\to N$ a continuos map is proper if (by definitios) $F^{-1}(K)$ is compact for any $K\subseteq N$ compact. Suppose now that $M,N$ are locally compact and Hausdorff. Then the following are equivalent: 1)$F$ is…
DDT
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Decide when product of discrete topologies is discrete itself

$\{X_{\alpha}\}_{\alpha\in \Lambda}$ be discrete topological spaces and $X=\prod_{i\in\Lambda} X_i.$ Then which of the following statements imply that the product topology on $X$ equals the discrete topology on $X\ ?$ $1.\Lambda \text{ is…
user118494
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Necessary and sufficient condition for the spaces $\mathbb R/A$ and $\mathbb R/B$ to be homeomorphic

Let $X$ be a topological space. Let $Y$ be a subset of $X$. We denote by $X/Y$ the quotient space of $X$ identifying any two elements of $Y$. Let $A$ and $B$ be two finite subsets of $\mathbb R$. Are $\mathbb R/A$ and $\mathbb R/B$ homeomorphic if…
Makoto Kato
  • 42,602
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The final segment topology of the naturals

We define the topological space $(\mathbb{N},\tau)$. The open sets of this topological spaces are the sets of the form $\{n+1,n+2,\dotsc\}$, for some $n\in\mathbb{N}$. This topology is called the final segment topology. My question is: is this…
Marios Gretsas
  • 21,906
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Constrution of a particular topological space

Let $n \in \mathbb N$ be a fixed natural number. Does there always exists a topological space $(X, \tau)$ such that $\vert \tau \vert=n$ ? I am interested in both cases when the cardinality of $X$ is finite and cardinality of $X$ in infinite? Its…
Dontknowanything
  • 2,810
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A question about spaces having a locally finite refinement

I've been reviewing questions from previous exams I've had in my courses, and I noticed a question that I completely missed. I was wondering if anyone could help me out on this one. Let $X$ be a topological space. Prove that every $\sigma$-discrete…
josh
  • 4,041
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Two retracts $A,B$ of $S^{2}$ with different relative location

Are there two retract subsets $A,B$ of $S^{2}$ with the following property: $A,B$ are homeomorphic but two pairs $(S^{2},A)$ and $(S^{2},B)$ are not homeomorphic. The same question for $S^{n}$
Ali Taghavi
  • 848
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Union of two locally compact space

Let $X=\mathbb{R^2}$ and let $$A=\{(0,0)\}\cup\{(x,y):x>0\}\subseteq X\;.$$ Notice that there is no compact set around $\{(0,0)\}$ because it will be open set. It looks like work for example of two locally compact space whose union are not locally…
Gob
  • 1,117
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Is this a sufficient condition for a subset of a topological space to be closed?

Let $X$ be a topological space, and let $\{U_i\}$ be an open cover. If $Y$ is subset of $X$ such that $Y\cap U_i$ is closed in $U_i$ (for each $i$), does this imply that $Y$ is closed in $X$?
M Davolo
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