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) 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

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)