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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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Set of accumulation points is closed for every subset

I was trying to prove the following statement: Let $(X,\mathscr{T})$ be a topological space. If the set of accumulation points of $\{x\}$ is closed for every $x\in X$, then the set of accumulation points of each subset of $X$ is closed. I've tried…
Yuki
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Basis to a manifold by coordinate balls

I was trying to show that "Every manifold has a basis of coordinate balls". My approach was like this: Given a manifold $M$, it's well-known that for every point $\mathbf{x}\in M$ there exists a neighborhood of $\mathbf{x}$ homeomorphic to an open…
Br09
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A continuous function and a closed graph

Suppose I would like to show that for 2 metric spaces $A,B$, a function $f:A\to B$ and the graph of $f$, $G=\{(a,f(a))\in A\times B|a\in A\}$ that $f$ is continuous IFF $G$ is closed (with an additional condition for the $(\Leftarrow)$ direction --…
Allen
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Countable product of first countable Spaces is first countable

Suppose $X_i$ are first countable Space , $X = \prod_{i=n}^{\infty} X_i $, Then $X$ is first countable Space in product topology. Is it first countable in Box topology. Is uncountable product of first countable space is first countable I have…
Struggler
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If every image under a continuous open map of a Hausdorff space is Hausdorff, show X has discrete topology.

Let $ ( X , \tau )$ be a $T_2$ topological space such that every open continuous image of $X$ is $T_2$. Show that $ \tau$ is the discrete topology on $X$. This is a question I have been thinking about and feel I am getting nowhere. The only thing…
Jmaff
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Base and subbase of a topology

I'm confused about subbases: the sub in the name suggests that a subbase $S$ is a subset of a base $B$ of a topology $T$. Can there be a topology $T$ such that it is generated by a subbase that is not a subset of a given base $B$ that generates…
Rudy the Reindeer
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How to show that the right half plane of $\mathbb{R}^2$ is homeomorphic to the open unit disk?

I am trying to show that $\{(x,y)\in\mathbb{R}^2:x>0\}$ is homeomorphic to the disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2<1\}$. I know that the disk is homeomorphic to the whole plane. A homeomorphism from $\mathbb{R}^2$ to the disk could be…
3x89g2
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Compact Topology and Coarsest Topology

Let $(E, \mathcal{T})$ be a compact Hausdorff space. It is well known that every topology $\mathcal{U}$ coarser than $\mathcal{T}$ such that $(E, \mathcal{U})$ is Hausdorff is equal to $\mathcal{T}$. Is the converse true? (that is: if $\mathcal{T}$…
francis-jamet
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Some topologies are more equal than others

On MathOverflow 5 years ago, I answered a question about Awfully sophisticated proofs for simple facts. I answered Fürstenberg's topological proof of the infinitude of primes. While the answer ultimately got several votes, there was criticism that…
user452
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Lindelöf'ize a space?

Although much weaker, Lindelöf is in the same spirit of compactness. It occurs to me whether there is such a thing like "Lindelöf'ization", since there are various kinds of compactification process for a space, that is, to embed a non-Lindelöf space…
Hui Yu
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Path on $\mathbb{R}^2\setminus \{x_1, x_2\}$

Consider plane $\mathbb{R}^2\setminus \{ x_1, x_2 \}$ without two points, and such closed path on this plane: (points on this picture are deleted points $x_1$ and $x_2$) Question: how to prove that this path isn't homotopic to zero? Appendix. As I…
Aspirin
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Subspace of Real-valued Functions

I've been looking at the product topology, and came across this question. Let X be the set of all real-valued functions which are zero outside of a countable subset of $\mathbb{R}$. Consider X as a subspace of $\mathbb{R}^{\mathbb{R}}$ with the…
Maria
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Can $\mathbb{R}$ be homeomorphic to some $X \times X$?

I got this question in my topology exam and I had no idea how to make it, topology can be so weird sometimes it's hard to imagine some spaces. My heart told me that it is not possible, but I do not know how to proceed on this. The easiest way seems…
Lotte
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If A is connected, is $\bar{A}$ connected?

If A is connected, is $\bar{A}$ connected? Here $\bar{A}$ is the closure of $A$. Here's my attempt at trying to prove this: Suppose that $\bar{A}$ is disconnected. Then, there exists open, disjoint, non empty subsets $U, V$ such that $U \cup V =…
user120227
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Extending a continuous map between the boundary of two cells.

I'm working in Lee's book on topological manifolds and have gotten stumped on the first question in chapter 5, the chapter on cell complexes. The problem is: Let $D$ and $D'$ be two closed cells not necessarily of the same dimension. Show that…
Mnifldz
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