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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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Examples of topological spaces satisfying certain properties

I have learnt that if a regular, second countable space is normal(Theorem 32.1 of Munkres's Topology), and that a regular, second countable space is metrizable(Urysohn metrization theorem, Theorem 34.1). So, I want to analyize these four properties…
Joshua Woo
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The topology of countable complements

Take the topology on $\Bbb R$, the real line, which is, $\tau=\{A\subseteq\Bbb R\mid\Bbb R\setminus A\text{ is countable}\}\cup \{\varnothing\}$. Can one find a convergence sequence in this topology? Because, Take a sequence $\{a_n\}$, And suppose…
topsi
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Topological proof on discrete topology where $X$ is infinite

How must I prove this problem? Let $X$ be an infinite set and let $T$ a topology in $X$ in which all infinite subsets of $X$ are open. Prove: $T$ is a discrete topology in $X$.
badet
  • 81
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Need a hint: prove that $[0, 1]$ and $(0, 1)$ are not homeomorphic

I need a hint: prove that $[0, 1]$ and $(0, 1)$ are not homeomorphic without referring to compactness. This is an exercise in a topology textbook, and it comes far earlier than compactness is discussed. So far my only idea is to show that a…
Aleksei Averchenko
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Boundaries of finite intersections and unions of sets

I apologize if this is a duplicate - I looked but didn't find one. This question is sort of a sanity check. Let $A$, $B$ be sets and define the boundaries $\partial A$ and $\partial B$ as usual. Is it true that both $\partial (A \cup B) \subseteq…
bryanj
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$S^1$ is homeomorphic to $[0, 1]/ \{0, 1\}$.

How can I able to show that $S^1$ is homeomorphic to $[0, 1]/ \{0, 1\}.$ I am learning quotient topology from K.D.Joshi's Introduction to GENERAL TOPOLOGY book. Where he mentioned that, "Let $f: X\to Y$ be a surjective function. Then $f$…
amita
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About Baire spaces

I'm having difficult to solve this: Determine whether or not $\mathbb{R}_l$ is a Baire space. I tried to aplly the following lemma: "X is a Baire space iff given any countable collection $\mathbb{U}_n$ of open sets in X, each of which is dense in X,…
User43029
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Do I miss something about a remark about topology in this lecture note?

I'm reading a lecture note in topology. There is a remark Remark 1.2. A quick induction shows that any finite intersection $U_{1} \cap \cdots \cap U_{k}$ of open sets is open. It is important to point out that it is in general not true that an…
Akira
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Show that the cantor set contains no isolated points

Consider an infinite sequence of subsets of the interval $[0,1]$ obtained in the following way; set $C_{0}=[0,1]$. $C_{1}$ is obtained by removing the middle open half of $C_{0}$, that is $$C_{1}= C_{0} -(1/4,3/4)=[0,1/4] \cup [3/4,1]$$ C2 is…
aloyce temba
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Is the Interior of Hyperbola homeomorphic to $\mathbb{R^{2}}$?

I know that the interior of the unit disk is homeomorphic to $\mathbb{R^{2}}$ by the mapping $(r,\theta)\to(\tan(\frac{r\pi}{2}),\theta)$. I am struggling to come up with a homeomorphic map from the interior of the hyperbola $x^{2}-y^{2}=1$ to…
Mr.Gandalf Sauron
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Pointwise convergence topology.

I would like to know if the following statement is true or false. Let $X$ be a compact Hausdorff space and an infinite set. If $X$ has a topology strictly weaker than the discrete topology, then the compact convergence topology is strictly finer…
student
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open and closed set in non-metric topology

I have just started on learning topology. And I saw this question in Amann and Escher's Analysis I (Exercise 10, page 247): Let X:={1,2,3,4,5} and $\mathcal{T}:=\{\emptyset, X, \{1\}, \{3, 4\}, \{1, 3, 4\}, \{2,3,4,5\}\}$ Determine the closure of…
outis
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What is the equivalent of generating a topology via a basis or subbasis for the closure operator definition of a topology?

What is the equivalent to generating a topology with a basis for closure operators satisfying the Kuratowski closure operators. Let $X$ be a topological space. Let $c : 2^X \to 2^X$ be its closure operator. The set of open sets definition of a…
Greg Nisbet
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Proving derived sets are closed

I am following a proof of the statement The derived set(the set of accumulation points) $A'$ of an arbitrary subset $A$ of $\mathbb{R}^2$ is closed. in a book. It starts with Let $q$ be a limit point of $A'$. If it is proved that q $\in A'$,…
Vinod
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Why are topological spaces defined in terms of open sets, and not in terms of connected open sets?

I'm starting to learn about point-set topology, and I find the definition of topological spaces and open sets to be very weird. Since we care about continuity in topology, it's odd that topological spaces are defined based on open sets, which can be…
Theemathas Chirananthavat
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