Talk:Topological space

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A word in the lead

It was written "In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods." Then an anonymous editor (IP 173.28.211.0) did "...that satisfies a set of axioms" with edit summary "The first sentence had a grammatical error- it was written 'satisfy' whereas it should be 'satisfies' since the subject is 'a set'." I reverted, with summary "no, these two sets satisfy, together: of points, and of neighborhoods"; he/she reverted with summary "The words 'together with', 'along with', 'as well as' and 'in addition to' do not make the subject plural".

Being not a native English speaker, I do not argue. But I feel that the meaning is now distorted. Indeed, the axioms relating points and neighbourhoods cannot be satisfied (nor violated) but just points (nor by just neighborhoods); it should be meant that they are satisfied by points and neighbourhoods (in concert); thus, by the set of points and the set of neighbourhoods. Let someone competent is English and mathematics decide, what to do. Boris Tsirelson (talk) 09:39, 15 June 2016 (UTC)

I think my modification may satisfy both.--Bill Cherowitzo (talk) 17:10, 15 June 2016 (UTC)

Nice. Boris Tsirelson (talk) 18:28, 15 June 2016 (UTC)

Zaunlen's addition

Below is a (copy of a) recent addition, removed by Deacon Vorbis as "There might be something worth saying here, but I'm not sure what you're trying to say -- it's worded rather confusingly". Let us think, how to say the "something worth saying". Boris Tsirelson (talk) 15:28, 2 August 2019 (UTC)

Definitions of the notion of a topological space can be obtained by considering structure definable in metric spaces (for example, the predicate of a subset being "open" or the relation of a set getting arbitrarily close to a point) and extracting some of the properties that hold in all metric spaces, thus getting a more general notion of space.

"may be defined as..."

The initial line contains:

> a topological space may be defined as

This is not as helpful to someone visiting that is looking to learn what the definition of "topological space" **is**. To contrast with something specific, the Iron ore page begins with

> Iron ores are rocks and minerals from which...

not "Iron ores may be defined as rocks and minerals..." I think the page would be improved if the sentence were changed to one of the following:

1. "a topological space is defined as..."

2. "There are several definitions for topological spaces. Under <certain mathy conditions>, they are defined as..."

3. Something else that makes it clear what they are, not what they are permitted to be called. — Preceding unsigned comment added by 198.45.19.113 (talk) 20:02, 11 March 2021 (UTC)

Good point. The manual of style recommends "a topological space is ..." (I do not remember where it is recommended). I have fixed this, and by the way I have added at the beginning an informal definition that is, in fact, an explanation of the purpose of the concept. D.Lazard (talk) 21:06, 11 March 2021 (UTC)

Definition of "a topology"

In this article, "a topology" was defined as a collection of open sets. This goes against the common practice in mathematics. The true fact is that a collection of open sets defines a topology, but there are many other ways to define a topology. Many common topologies are not defined by their closed sets (for example, Zariski topology, topology of uniform convergence, etc.). I have edited the article and the redirect Topology (structure)Topology (structure) for reflecting this. D.Lazard (talk) 11:45, 30 June 2022 (UTC)

Definition "any union (finite or infinite)"

The definition "any union (finite or infinite)" should specify "countable". 93.147.160.21 (talk) 11:01, 23 September 2022 (UTC)

There is no such restriction in the standard definition. D.Lazard (talk) 13:19, 23 September 2022 (UTC)

Definition via open sets

Is it just me or does it nowhere define a topological space? It only defines a topology TheGoatOfSparta (talk) 15:57, 23 September 2023 (UTC)

The second sentence of the article is More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, .... This clearly a definition. Nevetheless , this definition should better be recalled in section § Definitions. D.Lazard (talk) 17:23, 23 September 2023 (UTC)

What I meant to say is that under the "definition via open sets" section it doesn't define a topological space. I wasn't specific. TheGoatOfSparta (talk) 09:30, 25 September 2023 (UTC)

Definition via neighbourhoods

Can the neighbourhood M included in N be N? Does the neighbourhood M have to be the same for all neighbourhoods of x? TheGoatOfSparta (talk) 16:57, 23 September 2023 (UTC)

Split for Vietoris & Fell topologies

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

There is concensus to split Vietoris and Fell topologies to Vietoris topology and Fell topology respectively. Gramix13 (talk) 22:36, 9 June 2025 (UTC)

---

These seem to be way too specialized for a high-level, broad article like this, but a quick look through the lit indicates that they seem plenty notable enough to sustain articles of their own. Plonking this down here if anyone feels up to it. 35.139.154.158 (talk) 15:49, 19 June 2024 (UTC)

• Agreed, the Vietoris topology absolutely has a place on Wikipedia, but at the same time, it is a much more specialised concept than say, the quotient topology. It should perhaps be mentioned in this article, but it should have an article of its own. --193.28.84.180 (talk) 21:53, 12 January 2025 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

History?

The sentence -

In the 1930s, James Waddell Alexander II and Hassler Whitney first expressed the idea that a surface is a topological space that is locally like a Euclidean plane.

- is plainly false. Counterexamples from centuries (!) earlier exist.

The historically correct sentence is the following one:

In the 1920s and 1930s, James Waddell Alexander II and Hassler Whitney developed the manifold and cohomology formalisms that are in use today. 129.93.161.205 (talk) 14:55, 15 April 2025 (UTC)

What does it mean to "manipulate" a definition?

Is it customary to (temporarily) choose (or 'settle on') one definition of "a topology"?

As of the last time I checked, the last sentence of the lede paragraph of [the "Latest revision as of 00:12, 9 August 2025" version of] this article, said: [QUOTE:]

There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate.

and ... while I think I understand the 'ordinary' meaning of the verb "to manipulate", I am a little bit puzzled by how it is being used here ... at the end of that [above quoted] sentence.

(Caveat: I have tried to read "all or most" of this "Talk:" page, which is mostly pretty interesting. However, I have not read the entire article ... in fact, not even all of -- [more like, almost none of] -- the "Definitions" section. [That is, the section "Topological space#Definitions"].)

Definitions of [the word] "manipulate"

I checked https://www.onelook.com/?w=manipulate and, actually, it started out by giving me (in a box like an "infobox" or "nav box", on the right hand side of the page) a little table of definitions of the word "manipulate", from wikt:manipulate#Verb! (q.v.) ('which see'.)

Now, I realize that the use of the word "manipulate" here (as the last word of the first paragraph of this article) might not be [oops, it might not have a meaning that is] quite as "literal" as most of the "example" meanings shown there.

Perhaps ... people who know more than I do about how "math" people (or "mathematicians") talk about what they do, probably have more familiarity with (or, understanding about) how things go when people who have some reason to be using (or researching, or talking about) "Closeness (mathematics)" are doing what they do.

Aha ... maybe "manipulating" a definition, [kinda] means using it ... at least, temporarily

My guess is that, perhaps in a given instance of ... an effort to prove something, or an attempt to introduce -- or explain, or conjecture about -- some (new?) idea, that ... if they -- [meaning, a proverbial or garden variety "math" person] -- are dealing with anything in the area (or [sub-] "field") of Closeness (mathematics), they probably just choose (perhaps wisely) one definition [of what "a topology" is], among the several available (see the sub-sections of the "Topological space#Definitions" section!), and stick with that, at least temporarily.

Then, what happens next

(at least, until /slash "unless" they decide -- for some reason -- to switch to a different definition of what "a topology" is),

is -- [perhaps] -- that they find out whether or not the choice of a certain definition to use, makes a difference in some way (like, maybe it "does or does not" make it easier [or, harder!] to prove a certain lemma, or corollary, or main theorem ... or perhaps it just "does or does not" help in some other way ... such as, to simplify some already existing proof.

But if so, then ... I would not call that -- ("temporarily" adopting or "choosing" a certain definition, and "maybe" finding out, eventually, that doing so does [or does not!] make things more convenient) -- "manipulating" that definition. I could call it (something more like) finding the use of a given definition to be "more" or "less" convenient, for certain purposes.

Maybe it is time to change the last 7 words of that sentence?

See the first <blockquote>d sentence above in this section of this "Talk:" page. The last 7 words of that sentence are:

is easier than the others to manipulate.

One idea or "suggestion" might be, to just replace those last 7 words with [something more like]

is often more convenient to use than the others.

Just an idea. -- Mike Schwartz (talk) 07:23, 19 August 2025 (UTC)

It is clear that the author of this sentence is not a linguist, and possibly a native English. It is also clear that the intented meaning of "to manipulate" is "for mathematical manipulations". I agree with your suggestion of replacing "manipulate" with "use", but disagee with the replacement of "easier" with "convenient". I'll do the modification. D.Lazard (talk) 09:41, 19 August 2025 (UTC)

The repetition of "use" in the sentence seems not convenient. So, I replaced the end of the sentence with "which makes generally reasoning and proofs easier". D.Lazard (talk) 09:56, 19 August 2025 (UTC)

OK, thank you.

Instead of "which makes generally reasoning and proofs easier", I would probably recommend to say [something more like] "which often does a better job (than the other definitions), of facilitating reasoning and proofs."

However, maybe neither one of those "choices" of wording is the best choice. For now, I will leave it up to other Wikipedia readers who have opinions about this (perhaps including D.Lazard, or others) -- "if any!" -- to influence the consensus in some way. That could take the form of "being bold", or it might involve "chiming in" here.

Rock on. -- Mike Schwartz (talk) 07:55, 20 August 2025 (UTC)

I can't quite parse the current formulation, "which makes generally reasoning and proofs easier", it doesn't seem grammatically correct. Your suggestion seems better, but to be honest, I'm not sure that sentence should be there at all. The claim seems very subjective, and probably depends a lot on the specific theorems being proved. Some definitions make some things easier than others, but I doubt everything is easier using open sets. I'm going to remove this part, but feel free to revert if you disagree. Sheddow (talk) 23:07, 20 August 2025 (UTC)

Although "generally" does not mean "always", I agree with this removal. D.Lazard (talk) 08:52, 21 August 2025 (UTC)

Definition via neighborhoods doesn't match Hausdorff's

The section "Definition via neighborhoods" begins, "This axiomatization is due to Felix Hausdorff," but what follows is very different from Hausdorff's axioms. The citation given is to a book by Ronnie Brown that probably shouldn't be cited for anything except his own ideas about cubical foundations of homotopy theory and higher Van Kampen theorems.

Here's a link to Hausdorff's original book, page 213: https://archive.org/details/grundzgedermen00hausuoft/page/213/mode/1up

And a quick translation of his axioms, with slightly modernized notation:

• Every point $x$ belongs to at least one neighborhood $U_{x}$; every neighborhood $U_{x}$ contains the point $x$.

• If $U_{x}$, $V_{x}$ are two neighborhoods of the same point $x$, then there is a neighborhood $W_{x}$ that is a subset of both $(W_{x}\subseteq U_{x}\cap V_{x})$.

• If a point $y$ lies in $U_{x}$, then there is a neighborhood $U_{y}$ that is a subset of $U_{x}$ $(U_{w}\subseteq U_{x})$.

• For two different points $x$, $y$, there are two neighborhoods $U_{x}$, $U_{y}$ with no point in common $(U_{x}\cap U_{y}=\varnothing )$.

I can try to find time to edit this section, but if someone else wants to do it first, so much the better. Naddington (talk) 05:35, 14 November 2025 (UTC)

I checked out Brown's book, and to be fair to him, he's not claiming that these were due to Hausdorff; he just states them. Unfortunately, the text in our article seems to be lifted verbatim from Brown's book, making it a WP:COPYVIO, and we should really either get rid of it, or paraphrase it properly. It might be worth it to dive into the article history to see what happened here. Also, the axioms of Hausdorff that you've mentioned above describe (of course) a Hausdorff space, and not a general topological space as we generally define it today, so I'm not sure what's best here. –Deacon Vorbis (carbon • videos) 06:33, 14 November 2025 (UTC)

Here's a great source on the history of topology: Gregory H. Moore, "The emergence of open sets, closed sets, and limit points in analysis and topology," https://www.sciencedirect.com/science/article/pii/S0315086008000050 Naddington (talk) 17:33, 14 November 2025 (UTC)

Intuition for topological spaces

D.Lazard raised a point that the introduction using 'interiorness' is inaccurate for every space that can be considered a topological space using the weakest axioms since, in such spaces, all subsets, other than the space itself, if non-empty, may have empty interior.

It is unclear to me exactly how using interiorness to define topological spaces suggests that that interior must be non-empty

If someone were to ask me for intuition of what a topological space is, I would begin with the relation between a point and a set (interior, boundary, or exterior), using Euclidean space as a prototypical example.

I do not see how 'arbitrary closeness' is simpler, since the word 'arbitrary' carries the weight of several logical quantifiers, some of them quantifying neighbourhoods, a structure we have yet to define.

It is usual for an introduction to leave out minor nuances especially if they are against the intuition for a prototypical topological space. For instance, when one is first introduced to topological spaces, the mental picture is almost always an infinite, continuous, non-discrete space. This is in direct contradiction to the existence of topological spaces with a non-zero but finite number of points.

(Whether or not it is okay for an introduction to leave out any minor nuance if it interrupts the mental image of a topological space as a rubber sheet may be a point of discussion)

To me, the article Neighbourhood presents the notion of a topological space well enough for a beginner to grasp, and its introduction is closer to the use of interiorness over closeness.

A discussion for why the closeness definition is easier to understand is appreciated.

Mechanikin Mechanikin (talk) 13:49, 6 April 2026 (UTC)

The point is not which one of the common (equivalent) definitions of a topological space (through neighborhood, open sets an closed sets) is more intuitive. The point is that interiorness cannot be used to define the concept of a topological space. At least, there is no commonly used textbook that introduces topology through interiors, and this is a fundamental reason to not use such an approach, per WP:NOR D.Lazard (talk) 15:56, 6 April 2026 (UTC)

Axiomatic foundations of topological spaces#Definition via interior operators defines topological spaces using interiors. Not trying to use Wikipedia itself as a verifiable source, but it is possible to define it via interior operater Mechanikin (talk) 16:18, 6 April 2026 (UTC)