Talk:Topological space

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Needs more informtaion on history, motivation and applications. Also needs illustrations and references. Tompw 10:20, 15 October 2006 (UTC)

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Actually, X can't be found from T as the "union of its members", because how do you know the members (singletons?) are open, or that every element is actually in an open set. Nevertheless, closure under arbitrary unions and finite intersections is sufficient (and equivalent) and this is usually given as the definition in most textbooks, with the proviso that one often checks the empty set and the whole space in practice. The reason it is equivalent:

The empty set is the union of the empty subcollection (the union of the empty set is the empty set).

The whole space X is the intersection of the empty subcollection (the intersection of the empty set is the universe under consideration, in this case, X)

Okay, I give. I admit, I think too much like a set theorist. If you think of a topological space in terms of the abstract operations of union and intersection defined from P(P(X)) --> P(X), then including the "special cases" of T having the empty set and X is definitely redundant and counterproductive (which is worse..., having to remember that the empty set and X are always in the image of the union and intersection maps, or pretending that the empty set isn't really finite after all??) This is the case, e.g. when comparing systems of subsets closed under unions of sets of certain cardinalities and/or complements (i.e. families of closed sets, sigma-algebras, Borel sets, sets closed under unions/intersections of sets of bounded cardinality, etc.) But I realise again I'm thinking like a set theorist, not a normal person :-) I'll go and change it back. Revolver

During lunch break, I asked a mathematician next door for a definition of topological space and he said: as far as I recall, it is a set of sets that is closed under union and finite intersection and contains the empty set.

Why don't you drop X from the definition? It can be found from T as the union of its members.

That wouldn't be too useful. X plays an important and intuitive role, has to be defined anyway, and we should not minimalise definitions to the point they become hard to understand. --FvdP 15:49 Nov 28, 2002 (UTC)

Emphasize that there can be more than one topology on a set? We wouldn't want people thinking the standard topology on R is the only one.

Roughly speaking, open sets are thought of as neighborhoods of points; two points are "close together" if there are many open sets that contain both of them.

This statement doesn't sit too well with me. The basic relation in topology isn't between points and other points, it's between SETS and points. Trying to figure out "how many" open sets contain a pair of points won't give you much (if any) information about the topology at all. For instance, in Euclidean space, for every pair of points (distinct or not!) there exist the same number of open sets containing them, no matter what two points you choose. So, asking how many open sets contain both points doesn't tell you anything. But the relation between sets and points (an open set or neighbourhood containing a point is "close to the point") does (this way of thinking is basically the equivalent definition in terms of neighbourhood systems). Revolver

Why, when I click on the link to "Top" in the beginning of the article, does it take me to the EDIT page instead of the article?? Is this a problem on my end, my computer, or does this happen for everyone? The text of the article clearly gives the link as going to "Top", not to the edit page of "Top".

Just a question: What is this stuff practical for? I know of a lot of uses of the derivitives and integrals of calculus; uses of the volumes, angles, and surface areas of geometry and triginometry; and the MANY uses of simple algebra. What is topology good for? I just think it would be a good idea to try and point out practical uses of these things for people who aren't studying Algabraic Topology and such.

Quick explanation for the recent reversion: first, it is the complements of the open sets _in X_, not in the power set of X. The open sets are subsets of X, and their complements in X are closed; it doesn't even make sense to talk about their complements in the power set, because an open set isn't a subset of the power set (it's not a collection of subsets of X, it IS a subset of X). For the second change, it doesn't make sense to say, "the points of an open set are near each other", as I've said before, the fundamental nearness relation in topology is not between points and other points, it's between points and sets. It doesn't make any sense to say two points are "near" each other, because "nearness" is a relation between sets and points. In fact, if you define a topology using axioms for nearness relations (in essence, it turns out a set is near a point if and only if the set is a neighbourhood of that point (possibly not open)), then you go on to DEFINE a set to be open if and only if it is near each of its points, so the way I had worded it previously was precisely correct, in this sense. Revolver 03:02, 29 Jan 2004 (UTC)

It seems that some of the confusion may be from terminology. Actually, above, what I'm calling a "nearness relation" is usually called "neighbourhood systems" or "neighbourhood relation". In a certain sense, either relation can be thought of as "nearness" (somewhat near, or all near). Let me set out the 2 different way of doing this:

  • Nearness relation: Okay, this is NOT what I described above. I was confused (i.e. forgot) about what the conventional terminology is. A nearness relation is usually defined from a topology by defining a point to be near a set if and only if the point is in the closure of the set. Conversely, you can define a nearness relation as a relation between points and sets satisfying axioms that the former satisfied. For instance, every set is near each of its points (i.e. every set is contained in its closure), and the set of all points near a set A or a set B is equal to the set of all points near A ∪ B (closure of A ∪ B = closure A ∪ closure B), etc., etc. In this case, "near" means "somewhat near". And in this case, "open set" is defined as follows: a set A is open if no point of A is near the complement of A (i.e. if the complement of A is closed). Really, the nearness relation concept here is much more closely connected to the idea of closed sets than open sets.
  • Neighbourhood systems, or neighbourhood relation: THIS is what I was talking about above (erroneously calling a "nearness relation"). Here, given a topology, you define a neighbourhood relation by the way you would think it should be. Conversely, you can define a neighbourhood relation as a relation between points and sets satisfying axioms that the former satisfied. For instance, if a set A is a nbhd of a point x, then x is in A; also, if two sets are nbhds of a point x, then so is their intersection, etc., etc. In this case, "near" means "totally near", in the sense that a nbhd of a point is as close to it as possible (i.e. the complement is nowhere around). And in this case, "open set" is defined as follows: a set A is open if it is a nbhd of each of its points (this is what I really meant when I originally wrote "a set is open if it is near each of its points".

To me, each of these intuitively captures the idea of "nearness", but the former got the name attached to it. So, what should the article mean when it says "near"? We could choose the former, that agrees with current terminology, but seems kind of awkward without talking about closed sets ("a set A is open if no point of A is near the complement of A")...or we could choose the latter, going against terminology, but expressing the (more intuitive?) neighbourhood concept. Revolver 08:51, 29 Jan 2004 (UTC)

Roughly speaking, a topology is a way of specifying the concept of "nearness"; an open set is "near" each of its points.

This statement seems bogus to me as an intuitive explanation. An open set contains its points. That's like saying I'm "near" my liver. The original statement originated from the definition of convergence of sequences, and so was somewhat justified (ie, if the tail of a sequence lies in an open set, so does its limit). Nevertheless, I think in this case no intuitive explanation is better than a confusing one. An open set is, mainly, just an abstract concept described by the three rules given. Intuition about it should be acquired through examples, and we have plenty of those. Therefore I've removed this statement.

On another point, I think the article needs to mention bases and subbases, and I've added this too, although I'm not sure how good my placement is.

Derrick Coetzee 15:09, 30 Jan 2004 (UTC)

This statement seems bogus to me as an intuitive explanation. An open set contains its points. That's like saying I'm "near" my liver.
Agreed, it can be confusing as an intuitive statement. However, it DOES make sense to say a set is near all its points, if "near" is interpreted to mean "is a neighbourhood of". You just have to know how "near" is defined. If the intuitive meaning of "near" is too confusing, then maybe it shouldn't be there, but it does have a precise legit meaning in terms of neighbourhood systems.
ie, if the tail of a sequence lies in an open set, so does its limit
I'm not sure what this is intended to mean; in any case, it's not true...the sequence {1/n} lies in the open set (0, 1), but its limit 0 does not.
An open set is, mainly, just an abstract concept described by the three rules given. Intuition about it should be acquired through examples, and we have plenty of those.
An open set is a member of a topology, and a topology can be defined in any number of equivalent ways (the "neighbourhood" or "nearness" relation between 2 other ones). In many ways, the notion of "open set" is less intuitive than many of these other ways. Maybe we could replace this attempt at gaining intuition by ambiguous terms ("near") by giving the short standard introduction via metric spaces (i.e. show that the open sets in a metric space satisfy axioms of a topology)

Revolver 20:02, 30 Jan 2004 (UTC)

:ie, if the tail of a sequence lies in an open set, so does its limit

I think you mean "closed" instead of "open" here; in that case, it's certainly true. Revolver 20:06, 30 Jan 2004 (UTC)

I think maybe we should think about what we mean by the difference between intuition and motivation of mathematical definitions. I'm not quite sure they're the same thing. To me, intuition means a gaining of familiarity with a concept, a feeling of knowing it, comfortability with it, facility in handling it and proving things about it, making conjectures and guesses about it. As you say, this is almost certainly best acquired through examples (and problems). Motivation, on the other hand, to me means answering the question why do we care about this?, why is it useful, why on earth did anyone come up with it?, who cares?, what is it good for, what problems might it solve, how is it related to other areas of math, science, etc. It seems that motivation should preceed formal definition (or at least, preceed a formal exposition of the consequences of a definition, but gaining intuition by examples and problems should occur after the definition and after motivation. Sometimes the two overlap, because often examples are the best motivation. In any case, the def of a topology is probably too sophisticated to try to "sum up" in a short sentence or two. Revolver 20:14, 30 Jan 2004 (UTC)

Err, I screwed up that comment about sequences. I meant to refer to the definition that says every neighbourhood of the limit of a sequence contains a tail of it. Derrick Coetzee 23:02, 30 Jan 2004 (UTC)

[edit] weaker = coarser = smaller, stronger = finer = larger, not always the case?

In my experience as well as in all my topology books (e.g. Willard's General Topology, Steen's Counterexamples in Topology, as well as on Wikipedia (e.g. finer topology, weak topology, weak operator topology the terms weaker, coarser and smaller (and stronger. finer, larger) are used equivalently. Can anyone cite a source which reverses the meaning of any of these? If not I would like to modify the section on Relations between topologies to reflect this, as follows:

Given two toplologies T1 and T2 on the set X, T1 is said to be weaker ("coarser, smaller) than T2 iff T1 ⊆ T2. Paul August 02:51, Aug 15, 2004 (UTC)


[edit] Some questions

-What is the difference between the requirement for unions (or intersections) of collections of members of T to be in T and that for pairwise unions (intersections) to be in T? I think I might know the answer (?): although we can use the pairwise requirement to build the union of a sequence, viz. (((AuB)uC)uD)..... we can't do this for unions of uncountable collections of sets, so the 'collections' axiom is stronger because of this. Is this why they are different? If so a statement should be added to the definition to make this clear - I will do this if somebody confirms the above.

You are on the right track, but it's not just uncountable collections that you can't do this way - it's infinite collections.

-Following on from the above, why not use the reals as an example? This will make it clearer that the less advanced definition of an open set (any point in it has a neighbourhood which is also in it) is consistent with this one. This would go as follows: singletons in the reals are closed (under the neighbourhood definition), so is a finite or a countable union of such singltons, but an uncountable union of points _can_ result in an open set eg \bigcup\{\{x\}|x\in \{r\in R|1<r<2\}\}. Presumably the same can be said for uncountable intersections.

A countable union of singletons is not necessarily closed. For example, the set of rationals is a countable union of singletons, but it isn't a closed subset of the reals.

I'd really like it if someone could confirm these ideas. Also please give opinion on adding a couple of notes explaining these points - as someone reading this material for the first time my immediate reaction was "if I take a (finite) set of closed subsets of the reals, they satisfy these axioms (as the union of any collection will be closed and the intersection of any pair will also be closed, thus closed sets are open!". Why is this wrong (if it is)? Must topologies be uncountable collections of subsets? The axioms do not say so... SgtThroat 18:48, 19 Dec 2004 (UTC)

Topologies don't have to be uncountable, or even infinite. As for your example: it's possible to have a finite topology on the reals in which every open set is closed in the standard topology. Why do you see this as a problem? --Zundark 19:37, 19 Dec 2004 (UTC)

Excellent, thanks for your help - I think I see more clearly now.

I didn't (and still don't) know what "the standard topology" is. Perhaps my problem is that to me {x | 1 < x < 2} is open and not closed, wheras the reality is that it depends upon what the topology is; what I've learnt was all in the standard topology. Can I think of the standard topology as the set of all open (under the "traditional" definition of open) subsets of the reals? SgtThroat 02:22, 21 Dec 2004 (UTC)

Yes. When someone talks about topological properties of the reals it is always the standard topology (the Euclidean topology) that is intended, unless otherwise specified. The standard topology is the topology of the standard metric |x-y| and is also the order topology with respect to the standard order. Most of the time there is no need to consider other topologies on the reals. --Zundark 14:27, 21 Dec 2004 (UTC)
  • Can somebody expand an "empty collection"? I don't think "empty collection" is defined before it is used in: "Thus, since the union of the empty collection is the empty set." GoAwayStupidAI 19:01, 4 October 2006 (UTC)

[edit] Rubber sheets

Can anyone explain the relationship between topological spaces and the layman's intuitive notion of topology as "rubber-sheet geometry"? --P3d0 June 30, 2005 20:15 (UTC)

It depends on how flexible yet durable your rubber is :)

"Rubber-sheet geometry" comes from the homeomorphisms of topological spaces.--SurrealWarrior 02:44, 10 July 2005 (UTC)

Er... Ok. Is there a way to explain it such that a mathematically-inclined layman could understand it? --P3d0 01:24, July 11, 2005 (UTC)

Imagine an object made out of rubber. if you were to "squeeze" or "bend" or stretch" the object, some properties of the geometry of the object would change, for instance the distance between points. However other properties wouldn't. Roughly speaking those properties which remain invariant under such "transformations" (called homeomorphisms) are the objects "topological" properties. Thus a coffee cup and a torus are topologically the same, since one can be transformed into the other by bending and stretching without say "cutting" and/or "pasting". Paul August ☎ 05:23, July 11, 2005 (UTC)


Is there a way to metrisize the neighborhood systems of a topological space?--SurrealWarrior 8 July 2005 21:44 (UTC)

[edit] Topology versus topological space

The definition as it stands refers to a topology and not a topological space. I am advocating the use of the "erroneous" definition as it seems more logical for someone who is mentally sorting out the exact meaning of the relationship between sets in a topological space. Right now, "topology" is defined in the wikipedia as the study of topological spaces. As such, it falls on the "topological spaces" definition to give the criteria for the relation between these sets clearly.

The word topology refers both to the set of open sets in a topological space as well as to the branch of mathematics that studies topological spaces. It is, perhaps, an unfortunate overloading of terminology, but such is life. -- Fropuff 20:01, 7 September 2005 (UTC)
The word algebra is similarly overloaded. And I think there is a mathematical structure called a geometry. Now all we need is something called an analysis. -lethe talk 22:53, 26 January 2006 (UTC)

[edit] New To Advanced Math

Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as topological spaces, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006

First, let me mention a wikipedia convention. New posts go at the bottom of the page.
I like the book Topology by Munkres. Plenty of examples. If you find that book hard going, you might try the Schaum's Outline Series book Topology. Even more examples.
The key to understanding Topology is to keep track of the three levels of objects. First, "points", which make up the topological space. Second, "sets", which contain points. Some sets are open, some aren't. Then "families", which are sets of sets. A topology is a family and its elements are sets (one level down). A set is called "open" if it is an element of the topology. It helps you keep track of where you are if you name all points with lower case letters, all sets with capital letters, and all families with script letters.
Good luck. Rick Norwood 22:42, 26 January 2006 (UTC)
I also like Munkres. It helps to look at lots of examples of topological spaces that you already know from other math classes, such as Rn (a metric space), Q, closed bounded subspaces of Rn (one of the simplest examples of compact spaces), and also trivial cases such as the discrete topology and indiscrete topology. Deco 23:08, 26 January 2006 (UTC)

When studying continuity of real functions, or functions on Rn, one has to deal quite a lot with open sets. Those are sets that in some sense "do not contain their endpoint". Like the open intervals in R like a–ε<x<a+ε, or open balls of the form ||x – a||<ε. You should recognize those sets from the definition of a limit as well. One of the key properties of these sets is that every point in the set is contained in another set of the same type which itself is entirely contained within the first set. For example, 1.01 is contained in the set 0.95=1 – 0.05 < x < 1 + 0.05=1.05. So I can find another open interval inside the first interval (0.95,1.05) which also contains 1.01. For example, the interval (1.005, 1.015) fits the bill perfectly. Because the set doesn't include its endpoints, we can get as close to the endpoint as we want, and still stay inside the set. This does not work for closed intervals. For example, the point x = 1 of the closed interval [0,1]; any interval containing 1 must pass outside [0,1]. There are no intervals around 1 which are entirely inside that set. This property of open sets is what characterizes it. To say it again: every point in an open set is contained in an open subset of the open set.

Now, going into the 20th century, all of mathematics became more formal. Everything had to be defined by axioms, all branches of mathematics had to be axiomatized. Before this, groups were things that permuted numbers, like the symmetry group {id,(12)}, or were symmetries of objects, like the dihedral group which rotates a polygon so that vertices go to vertices, or matrix groups, like the set of rigid rotations of Euclidean space. After the axiomatization, a group was allowed to be anything at all satisfying the three axioms of identity, inverse, and associativity. That is to say, elements of a group were no longer geometrical or combinatorial operations. Now they became abstract symbols with no meaning whatsoever, except that they followed certain properties. People who were familiar with the old-fashioned symmetry groups chose the axioms carefully to make sure that the axioms caught all the properties of those groups. With the new abstract way of thinking, all kinds of new groups were discovered. The capacity to think of groups more generally was born, and it became possible to at least ask the question "can we list all groups?" (the answer is "no", by the way). But the price of this generality is abstraction. Multiplication is no longer the same thing you were taught in school. It can be any set, any operation, as long as it's associative.

So topology. It wasn't exactly the same as with groups, since topology wasn't born until the 20th century or so, but the axioms of a topological space are meant to catch the essential property that I mentioned above: every point in an open set is contained in an open subset. The price for the abstract definition is that open sets are no longer the open intervals you learned in school. Now they can be any collection of sets that is closed under union. But you should keep in mind that defining characteristic of open sets if you find the definition too confusing. And now here's your exercise: prove that if U is an open set (according to the abstract definition) if and only if every point in U is contained in an open set contained in U. -lethe talk 23:33, 26 January 2006 (UTC)

[edit] remove bullet points

I've removed the bullet points, done a minor rewrite, and removed a few errors (for example the statement that the open sets in a finite product topology are all products of open sets).

I just asked a graph theory expert what a "linear graph" is and she didn't know. Whoever put this in needs to provide a definiton. Rick Norwood 21:08, 8 February 2006 (UTC)

[edit] Number of topologies

Can you please tell me how many topologies may be defined on a space X which is composed of |X| numbers?

And, what about the simpler problem? What is the cardinal number of all the topologies on a given inifinite space X? |P(X)|? |P(P(X))|?

Thanks, 80.178.181.114 13:52, 9 April 2006 (UTC)

Interesting question. Rick Norwood 14:43, 9 April 2006 (UTC)
If X is infinite, then there are |P(P(X))| topologies on X, where P(X) is the power set of X. I don't think there's a simple formula if X is finite. --Zundark 16:09, 9 April 2006 (UTC)
Can you prove the first statement? 80.178.241.176 19:42, 15 April 2006 (UTC)
It follows immediately from the fact that there are |P(P(X))| ultrafilters on X, since each ultrafilter becomes a topology if you add the empty set. --Zundark 16:52, 22 April 2006 (UTC)

[edit] Typo

"If Γ is an ordinal number, then the set [0, Γ] then a topology is generated if the basic open sets are taken to be the intervals (a, b], where a and b are elements of Γ."; I don't know what this means so I won't try fixing it, but it's grammatically incorrect.

I can't fix it either. I think the standard topology to consider on ordinals is the order topology, whose topology has basis (a,b), whereas the article seems to prefer a basis given by (a,b]. These topologies are different. -lethe talk + 00:50, 19 May 2006 (UTC)
I've changed it to what I think it ought to say. The grammar is hopefully correct, but of course the example includes open intervals now, instead of half-open intervals. -lethe talk + 01:48, 19 May 2006 (UTC)

[edit] pairwise vs finite intersections

I've corrected a couple of inaccuracies in the last paragraph of the "Definition" section. Formerly it said that being closed under pairwise intersections and being closed under finite intersections, are equivalent as axioms. They aren't (since finite includes zero). Also axiom 1 is not redundant, since if you only require that T to be closed under pairwise intersections, you need to require X is in T. Axiom 1 is redundant if you require closure under finite intersections.

That paragraph now reads:

By induction, the intersection of any finite collection of open sets is open. Thus, an equivalent definition can be given by changing the third axiom to require that the topology be closed under all finite intersections. This has the benefit that we need not explicitly require that X be in T, since the empty intersection is (by convention) X. Similarly, we can conclude that the empty set is in T by using axiom 2 and taking a union over the empty collection.

I propose replacing it with:

By induction, the intersection of any finite collection of open sets is open. Thus, since the union of the empty collection is the empty set, and the intersection of the empty collection is (by convention) X, an equivalent definition can be given by requiring that a topology be closed under unions and finite intersections.

I think this reads better. Any objections? Paul August ☎ 16:55, 15 June 2006 (UTC)

No objection from me. -lethe talk + 17:34, 15 June 2006 (UTC)
If we drop the empty set and whole space axiom, can we say that the empty set is a topology? -lethe talk + 03:50, 16 June 2006 (UTC)
No, the empty collection of subsets of X is not a topology, since it satisfies none of the axioms. Paul August ☎ 15:22, 16 June 2006 (UTC)
No? It seems to me that "the union of any collection of elements is again an element" is vacuously satisfied by the empty collection, and similarly for intersection. -lethe talk + 15:36, 16 June 2006 (UTC)
No. For "the union of any collection of elements is again an element" to be vacuously true for the empty set, the empty set would have to have no subsets to take a union over, but of course the empty set has a subset, namely the empty set itself, and the union of the empty set is not an element of the empty set. Make sense?Paul August ☎ 19:34, 16 June 2006 (UTC)
I'm a little dizzy from the argument, but yes, I think it makes sense. And it's reassuring to hear, because otherwise I would conclude that in fact these axioms were not equivalent. -lethe talk + 20:24, 16 June 2006 (UTC)

I am not sure that this belongs in the article. It is of technical interest to mathematicians, but all of the introductory topology textbooks I've seen list the four axioms. That it can be done with fewer axioms is interesting, but I'm not sure it helps in understanding what topology is all about. Yes, any union over the empty set is empty, any intersection over the empty set is the entire space, so we could have just two axioms, or even just one axiom: A topology is a family of sets closed under all unions and under finite intersections. But minimalism is not necessarily a good thing in an encyclopedia. Rick Norwood 13:10, 16 June 2006 (UTC)

I kinda like it. Do we really have to lose it? -lethe talk + 13:49, 16 June 2006 (UTC)

How about if it goes further down in the article? Rick Norwood 16:01, 16 June 2006 (UTC)

Well we want the first sentence at least I think. I agree that the second sentence is something of a side issue or mathematical curiosity. But it is sometimes given as the definition (e.g. Bourbaki) and some people will have that as the definition of a topology and might be confused if they don't see it. In fact, the reason I made the recent edits that I have, is because some anon added this definition in the "Equivalent definitions" section, without apparently realizing that it was already covered in the "Definition" section.
In any case, for now, I've gone ahead and replaced the existing paragraph with my proposed one since no one has said they think the current one is better.
I agree with Rick's remark about minimalism not necessarily being a good thing. And in fact our current definition already perhaps suffers from minimalism. Most often the intersection axiom is given as requiring closure under finite intersections. Which I might prefer. If we did that then we could handle Bourbaki's definition by simply mentioning that the first axiom was technically redundant. We would then however, I think want to mention somewhere that closure under pairwise intersection was sufficient once it is know that the entire space is in the topology. As a practical matter, when showing that a collection of subsets T of X is a topology, the usual procedure is to show four things: that T contains, ∅, X, non-empty unions of elements of T, and pairwise intersections of elements of T.
Paul August ☎ 16:43, 16 June 2006 (UTC)
I wrote the above before I saw Rick's last comment. Further down might be ok, but before moving it let's discuss my last paracgraph above. Paul August ☎ 16:47, 16 June 2006 (UTC)
What happened to your little telephone? -lethe talk + 16:50, 16 June 2006 (UTC) bad browser
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