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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 the Discrete Topology on $X$ the Only One Containing All Infinite Subsets of $X$?

Prove or find counterexamples. Let $X$ be an infinite set and $T$ be a topology on $X$. If $T$ contains every infinite subset of $X$, then $T$ is the discrete topology.
M.Sina
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Why the definition of topology is what it is?

I recently attended an interview for PhD. One of panel members asked me definition of topology which I answered and next question was why the definion supposed to be that way. I explained through an example. Is it right to explain definition…
Rahul Raju Pattar
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Countable number of arcs between two intervals

First consider the two intervals $I_0=\{0\}\times [0,1]$ and $I_1=\{1\}\times [0,1]$ in the plane. Suppose that, for each $n\in\omega=\{0,1,2,...\}$, $A_n$ is an arc (a homeomorphic image of $[0,1]$) contained in $\mathbb [0,1]^2 \setminus…
Forever Mozart
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Request for gentle explanation of defining a topology with its universal property

I am reading a book which says that whenever we can define a topology by saying "its the largest topology which satisfy $p$" then it is possible to define the same topology by saying it is the "smallest topology which satisfy $q$". Why is that? …
Hooman
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Can real numbers be dense in complex numbers for some topology?

I am looking for a topology such that R is dense in C . I was thinking I can construct a surjective continuous function f : R → C such that the image of Q is R.
Ißra Jörg
  • 115
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The unit ball $S^n$ is homeomorphic to the one-point compactification of $R^n$

I saw many different proofs in the community of the statement in the title. And I want to demonstrate my proof of the general case. I have tried to show the one-point compactification is homeomorphic to $S^n$ directly, but encountered difficulty in…
Oscar LIU
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How can I tell whether this set is closed or open?

the question I'm having trouble with is this: $$A:=\bigcup_{n=1}^{\infty}\left({\left[\frac{1}{n+1},\frac{1}{n}\right) \times \left(0,n\right)}\right).$$ In the Euclidean space $\mathbb{R}^2$, is the subset $A$ either closed or open? I drew a…
Peter
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A Compact, Hausdorff topological space has finitely many components?

I'm not sure if this is a true statement or not, but due to the compactness, it seems like it should be. My attempt at a proof involves supposing it has infinitely many components, and then taking a sequence such that each element in the sequence is…
notsure
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A curious compactness confusion: space filling curves in the hilbert cube that contradict a bona fide theorem?

Now, I may have only slept two hours last night and would currently struggle to discern a 'proof' by induction of FLT from a piece of genuine mathematics, but that doesn't stop mathematics from bugging me. At present I am puzzled by something I saw…
Tom Boardman
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$(0,1)\cap \mathbb{Q}$ is not compact?

I hope you would read my approach. First, choose an increasing sequence, $\{a_n\}$ of irrational numbers, the limit of which is $1$. Then, the open covering, $\{(0,a_1),(a_1,a_2),(a_2,a_3),\ldots\}$ has no finite subcovering. Therefore, $(0,1)\cap…
user406753
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A property of an open cover

Let $\{A_i\}_{i\in S}$ be an open cover for a regular topological space $X$. The family $\{A_i\}_{i\in S}$ is called locally finite if for every point $x\in X$ there exists a neighborhood $U$ of $x$ such that the set $\{s\in S : U \cap A_s\neq…
Deroty
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Is the Hausdorff dimension invariant under homeomorphisms?

Recently I was thinking extensively about dimensions of a spaces and ascertained that algebraic or topological definitions of dimensions could hardly be not an integer. So I'm curious whether the Hausdorff or Minkowski dimensions invariant under…
HyJu
  • 701
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Proof that a $T_1$ Space has a locally finite basis iff it is discrete

I'm working my way through Munkres, and I'm having difficulty with exercise 3 of section 40 on the Nagata-Smirnov Metrization Theorem. Many spaces have countable bases; but no $T_1$ space has a locally finite basis unless it is discrete. Prove this…
Itserpol
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$S^3$ $\cong$ $D^2\times S^1\bigsqcup_{S^1\times S^1} S^1\times D^2$?

Is $S^3$ homeomorphic to $D^2\times S^1\bigsqcup_{S^1\times S^1} S^1\times D^2$ ? Here $D^2$ denotes the closed 2-dimensional unit disk. If it is, how to prove it?
ougao
  • 3,671
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Prob. 7, Sec. 20 in Munkres' TOPOLOGY, 2nd ed: The coordinate-wise linear self-map of $\mathbb{R}^\omega$

Here's Prob. 7, Sec. 20 in the book Topology by James R. Munkres, 2nd edition: Consider the map $h \colon \mathbb{R}^\omega \to \mathbb{R}^\omega$ defined in Exercise 8 of Sec. 19; give $\mathbb{R}^\omega$ the uniform topology. Under what…
Saaqib Mahmood
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