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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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Is a hollow sphere topologically equivalent to a torus?

I'm not very well-versed in topology, but I know the basic concept of topological equivalence can be approximated by "counting the holes" in objects, in the sense that a sphere is different from a torus, but a torus is the same as a coffee mug…
sonaxaton
  • 121
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Function that is constant on connected components but not locally constant

The Wikipedia page for locally constant function says that a locally constant function is constant on each connected component, but that the converse only holds if the space is locally connected. What would be an example of a function that is…
Anakhand
  • 2,572
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Questions about open sets in ${\mathbb R}$

Consider the following problem: Let ${\mathbb Q} \subset A\subset {\mathbb R}$, which of the following must be true? A. If $A$ is open, then $A={\mathbb R}$ B. If $A$ is closed, then $A={\mathbb R}$ Since $\overline{\mathbb Q}={\mathbb R}$, one can…
user9464
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3 answers

Continuity on a union

Let $X= \bigcup X_n$ be a countable union of subsets $X_1\subset X_2\subset \dots$ and let $X$ and $Y$ be topological spaces. Given a function $f\colon X\to Y$ such that $f|_{X_n}$ is continuous, is it necessarily true, that $f$ is continuous on…
Peter Patzt
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Weaker/Stronger Topologies and Compact/Hausdorff Spaces

In my topology lecture notes, I have written: "By considering the identity map between different spaces with the same underlying set, it follows that for a compact, Hausdorff space: $\bullet$ any weaker topology is compact, but not…
lokodiz
  • 2,340
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Prove that if X is second-countable and every compact subset of X is closed, then X is Hausdorff.

Question: Prove that if $X$ is second-countable and every compact subset of $X$ is closed, then $X$ is Hausdorff. I know that the second-countability of $X$ is what will make the proof work at some point, since if you remove that from the…
WayMoreQuestionsThanAnswers
  • 265
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Convergence in the product topology iff mappings converge

Let $x_1,x_2,\ldots$ be a sequence of points of the product space $\prod X_\alpha$. Show that this sequence converges to a point $x$ if and only if the sequence $\pi_\alpha(x_1),\pi_\alpha(x_2),\ldots$ converges to $\pi_\alpha(x)$ for each…
emka
  • 6,494
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Is this a sufficient condition for two spaces to be homeomorphic; proof or counter example please.

Let $X$ and $Y$ be topological spaces. let $f: X \to Y$ and $g: Y \to X$. Assume that both $f$ and $g$ are continuous bijections. Can we say that $X$ and $Y$ are homeomorphic? If not are there assumptions we can place on the spaces so we do know…
Elijah Fender
  • 101
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how to find the unique smallest topology?

I have come across to the following question : Let $\mathscr{T}_\alpha$ be a family of topologies on $ X$ . Show that there is a unique smallest topology on $X$ containing all the collections $\mathscr{T}_\alpha$…
leopard
  • 407
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Non-trivial topology where only open sets are closed

For example, on $\mathbb{R}$ there exists trivial topology which contains only $\mathbb{R}$ and $\emptyset$ and in that topology all open sets are closed and all closed sets are open. Question. Does there exist non-trivial topology on $\mathbb{R}$…
Thom
  • 796
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Definition of Topological Embedding

The following definition of topological embeddings is given in Introduction to Topological Manifolds by John M. Lee. Definition. An injective continuous map that is a homeomorphism onto its image (in the subspace topology) is called a topological…
Koda
  • 1,196
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4 answers

Understanding two similar definitions: Fréchet-Urysohn space and sequential space

Here are the definitions: Fréchet-Urysohn space: A topological space $ X $ where for every $ A \subseteq X $ and every $ x \in \text{cl}(A) $, there exists a sequence $ (x_{n})_{n \in \mathbb{N}} $ in $ A $ converging to $ x $. Sequential space: A…
Paul
  • 20,553
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Why isn't a continuous bijection from a locally compact space to a Hausdorff space an homeomorphism?

I know that if $f : X \rightarrow Y$ is a continuous bijection from a compact space $X$ to a Hausdorff space $Y$, then $f$ is an homeomorphism. So I was thinking that if we relax the assumption $X$ compact to $X$ locally compact, it should be true…
Desura
  • 2,011
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2 answers

A topological space with cardinality strictly less than its weight?

My book states that if $X$ is compact, then $w(X) \leq |X|$. This leads me to wonder, is there a nice example of a (non-compact) topological space where $w(x) > |X|$ holds? The weight of a topological space is the smallest cardinality of a basis…
theQman
  • 1,097
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Are any of these notions of "k-space" equivalent if $X$ is not assumed weakly Hausdorff?

Is there a modern generally accepted answer regarding the notions of k-space or compactly generated space? For example there are currently at least 3 formally distinct notions of k-space in wide circulation: In Kelley's General Topology, $X$ is a…
Paul Fabel
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