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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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Union of $x$-axis and $y$-axis is not a manifold

Show that the union $X$ of the $x$-axis and the $y$-axis in $\mathbb{R}^2$ is not a manifold. Is the following a valid way of arguing? Suppose $X$ were a manifold. Then there would be a nbhd $U$ of the origin in $X$ that is homeomorphic to…
iwriteonbananas
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(Kelley's General Topology) Exercise G chapter 1.

I am finding difficultes in solving the following exercise written on the Kelley's book as in the title. Could anyone help me? Thanks in advance. If $A$ is dense in a topological space and $U$ is open, then $U \subseteq \overline{(A \cap U)}$.
Biagio
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Is second-countability invariant under homotopy equivalence?

I am wondering if second-countability is invariant under homotopy equivalence. If I had to guess I would say so. Intuitively, if we have a countable basis of a space, and then stretch, contract, bend the space, I don't see how we could get an…
iwriteonbananas
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A problem in topology relating to the finite intersection property

This is a problem from Munkres' Topology Exercise 37.1 (c) Let $X$ be a space. Let $\mathscr{D}$ be a collection of subsets of $X$ that is maximal with respect to the finite intersection property. (c) Show that if $X$ satistifes the T1 axiom, there…
nomadicmathematician
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A space is completely regular if and only if it is homeomorphic to a subspace of $[0,1]^ J$ for some $J$

This is a theorem from Topology by James Munkres. Theorem 34.3 A space is completely regular if and only if it is homeomorphic to a subspace of $[0,1]^ J$ for some $J$ The book merely states that this theorem is an immediate corollary of the fact…
nomadicmathematician
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Show that the set of all complex numbers $z$ such that $|z| \leq 1$ is closed?

I'm working through Rudin's "Principles of Mathematical Analysis" on my own, so I don't want the full answer. I'm only looking for a hint on this problem. Rudin states without proof that the set $X = \{z \ \text{complex}: |z| \lt 1\}$ is not closed.…
M T
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Homeomorphic closed subspaces.

Let $X$ be an arbitrary topological space, and $U,V\subseteq X$ two subspaces of $X$ such that $U\cong V$ ($U$ and $V$ are homeomorphic) with respect the subspace topology of $X$. I know examples where $U$ is closed in $X$ and $V$ is not. Is there…
Asupollo
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Dense Open Sets and Cocountability

Are all dense open sets in R cocountable? That is, are all dense open sets in R such that their complements are at most countable? It would seem like they must be since the closed sets that are uncountable are all intervals, so their complements are…
David
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Is a set open in a product of spaces if all its segments are open in their factors?

Let $X$ and $Y$ be topological spaces and let be $U ⊂ X × Y$ such that $$∀(x,y) ∈ X × Y \colon \quad \begin{aligned}•~~\mathrm{incl.}_{(–,y)}^{-1}(U) ~\text{is open in $X$,}\\•~~\mathrm{incl.}_{(x,–)}^{-1}(U)~\text{is open in…
k.stm
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Example of a $T_1$ space that does not have the property that every compact subspace of $X$ is closed.

Let $X$ denote a topological space. Then each condition in the following list implies the next. $X$ is $T_2$ Every compact subspace of $X$ is closed. $X$ is $T_1$. I know that 2 does not imply 1 (see here). I'm also guessing that 3 does not imply…
goblin GONE
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Continuous map between spheres

Possible Duplicate: Continuous map $\mathbb{S}^n\to \mathbb{S}^m$ Why is every continuous function $f:\mathbb{S}^n\to\mathbb{S}^m,$ for $n
Aspirin
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Union of infinitely many closed sets

If $(K_i)_{i \in \mathbb{N}}$ is a sequence of closed sets in $\mathbb{R}^3$, then the union of these sets $\bigcup_{i=1}^\infty K_i = K_1 \cup K_2 \cup ... $ is also closed. My idea: ($\bigcup_{i=1}^\infty K_i)^C = \bigcap_{i=1}^\infty…
fear.xD
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A proof of a small topological lemma

I just stumbled upon a proof of topological lemma that I don't understand: it would be great if anyone could give me some advices. To be blunt, I am convinced that the proof does work but to me it seems like the author quotes some results that are…
user160738
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Homeomorphism between plane with different topologies

How would you show that spaces $(\mathbb{R^2},\cal{T}_r)$ and $(\mathbb{R}^2,\cal{T}_b)$, where $\cal{T}_r$ is a topology generated by jungle river metric (here) and equivalently $\cal{T}_b$ is generated by the British Rail metric (3.15 second one),…
Jules
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Is $\mathbb{R}^{[0,1]}$ separable?

I was trying to disprove (or also prove) whether $\mathbb{R}^{[0,1]}$ is separable. My intuition tells me it's a disprove. I thought perhaps proving that $\mathbb{R}^{[0,1]}$ is sequentially compact will assist? Or maybe that won't help? And my…
Zhan I.s.
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