Talk:Continuous function (topology)

Balance of formality and accessibility

I have rewritten the introduction (again) in a way which is appropriate for an article on topology, but emphasising that the definition for metric spaces is exactly the same as the topological one. Using neighbourhoods rather than open sets is far clearer, because continuity is essentially about what happens near individual points. The reason the very abstract open set formalisation works is that an open set is simply any set that contains a neighbourhood of every one of its points, so continuity in terms of open sets is about continuity in parallel at every one of the points in the sets.

The reason I object to a too informal introduction is that it is easy to give a false idea of what continuity is. It is easy to define a function f on the plane which is discontinuous, but for which limits of sequences are preserved (i.e. any sequence {x_n}with a limit L is mapped to {f(x_n)} with a limit f(L). Anyone reading an informal definition about nearness could easily assume it would be enough for limits of sequences to be preserved.

There is still some stuff late in this article which I think is misleading because it ignores the fact that functions don't have to be injective. Elroch 01:05, 27 April 2006 (UTC)

Are you sure about this claim that a discontinuous function can still preserve limits of sequences? I am under the impression that in any first countable space, a function is continuous if and only if it preserves limits of sequences. In fact, I added material asserting that fact to the article a few weeks ago. -lethe ^{talk +} 01:29, 27 April 2006 (UTC)

You are right. I was thinking of the function:

$f(re^{i\theta })={\frac {r}{\theta }}e^{i\theta }\ldots (0<\theta \leq 2\pi )\;$

and hastily assumed it preserved limits of sequences at the origin. But it doesn't, and your general result is one I recall to be true. However my observation is true for spaces that are not first countable, which is relevant for a general topological concept. Elroch 11:52, 27 April 2006 (UTC)

Yes, indeed, your observation is still true for non-first countable spaces. Then you have the slightly generalized version which says that a function is continuous if and only if it preserves limits of nets (generalized sequences). Many people define continuity this way, in fact, so this view should be represented in the article. -lethe ^{talk +} 20:20, 27 April 2006 (UTC)

I favour sacrificing formality for accessibility especially in the intro, but I think yours is better than the "nearness is measured..." wording for both qualities. Thanks.

As for misleading stuff later in the article, if you are referring to the statement that any neighbourhood V of f(x) contains an image f(U) of a neighbourhood U of x, rather than the statement that the inverse image of a neighourhood of f(x) contains a neighbourhood of x: my understanding is that the former is also correct even when f is not injective. Am I mistaken? -Dan 14:18, 27 April 2006 (UTC)

The thing I was referring to was about the topology induced by a continuous function. I elaborated this by explaining the quotient topology and dealing with functions that are not necessarily surjective. Elroch 15:03, 27 April 2006 (UTC)

The stuff about the quotient topology is a bit misleading. Every function from a set determines a final topology. Every function also determines an equivalence relation (x~y iff f(x)=f(y)). If the function is surjective, it's true that the quotient space and the final topology on the codomain are homeomorphic. I wouldn't say that they're equal though; the quotient space is a space of equivalence classes, while a general codomain need not be. I've cleaned up the section, and also included the dual case, but now we've got two long paragraphs of stuff that I'm not sure belongs in this article. -lethe ^{talk +} 21:34, 27 April 2006 (UTC)

Other Comments

I think that it would be helpful if there were some (if they're too complicated then don't add them) proofs that the definitions were equivalent because a lot of the ideas are quite different and the proofs would help to show how they're connected. Trogsworth 16:53, 27 September 2007 (UTC)

Closeness relation definition

The definition using a closeness relation contains a link to closeness relations. However, in the linked page, closeness is only defined between a point and a set or between two sets; it is undefined what it means that two points are close. -- dnjansen 6 October 2008 —Preceding unsigned comment added by 131.174.42.93 (talk) 15:48, 6 October 2008 (UTC)

Topological definition of continous function

Looking through the history you will see that I attempted to make a change in line with pg 4 of :

http://www.dpmms.cam.ac.uk/site2002/Teaching/IB/MetricTopologicalSpaces/2005-2006/L1topspaces.pdf

Namely I changed :

suppose we have a function $f:X\rightarrow Y$, where X and Y are topological spaces.

To :

suppose we have a function f : X → Y between two topological spaces {X,T_X} and {Y,T_Y}.

The motivation for the initial modification being that the map f is defined between the sets X and Y, as opposed to only being defined between the topologies T_X and T_Y.

However the change was undone. Would it be possible to get some thoughts on these two alternative expressions, please?

Thanks —Preceding unsigned comment added by Arjun r acharya (talk • contribs) 11:17, 1 February 2009 (UTC)

I have now put the information above in a footnote. —Preceding unsigned comment added by 212.183.134.128 (talk) 15:52, 1 February 2009 (UTC)

continuous map?

The title of this page may be a bit of a neologism. Continuous morphisms between topological spaces are usually referred to as maps, not functions. Tkuvho (talk) 18:05, 4 March 2010 (UTC)

I notice that Acharya similarly used the term "map" in his comment on this page a year ago. I suggest we move the page to "continuous map (topology)". Tkuvho (talk) 05:41, 5 March 2010 (UTC)

I further notice that some of the pages linking to this page in fact use the expression "continuous map" with a redirect of type "continuous function (topology)|continuous map". Tkuvho (talk) 05:44, 5 March 2010 (UTC)

Requested move

{{movereq|Continuous map}} Continuous function (topology) → Continuous map

• The term "Continuous map" is the standard term for the concept described here. I am unable to move the page as continuous map already exists and is redirected to continuous function. Tkuvho (talk) 08:50, 5 March 2010 (UTC)
• Keep the disambig. To most people, "map" is not topology but what people look at to find where places are or to find their way around. Anthony Appleyard (talk) 09:58, 5 March 2010 (UTC)

Your remark is not really consistent with the fact that continuous map currently redirects to continuous function. Certainly "maps" have more to do with finding your way around than with topology, but the current redirect is an indication that "continuous map" is a standard term in mathematics that refers to a certain generalisation of a continuous function. The generalisation described in this page is almost never referred to as a "continuous function", and therefore the name of the page is inappropriate. If you check the pages that link to this page, you will see that most of them actually use the term "continuous map", rather than "continuous function". Tkuvho (talk) 20:00, 6 March 2010 (UTC)

I don't think we should have two different articles, one entitled "continuous map" and the other entitled "continuous function". That will only lead to confusion, especially because many people use the words "map" and "function" as complete synonyms.

I would prefer to simply merge the content from continuous function (topology) to continuous function. The articles together are under 40kb, so there is no problem with excessive length. The combined article could still cover continuity on the real line first, and then cover topological continuity lower down. So a basic reader could read the top part, and a more advanced one could continue down to the more advanced material. — Carl (CBM · talk) 12:46, 10 March 2010 (UTC)

• Well, "continuous function" and "continuous map" are used pretty much interchangebly in the literature, sometimes within the same text. (See Alan Hatcher's Algebraic Geometry for a prime example.) Sometimes one of them is given extra meaning. For example functions being maps taking values in C or R, or a function being a continuous map (doesn't Bredon's book do the latter?). Anyway, it is hard to tell which is more common. (I also agree with Carl) TimothyRias (talk) 12:54, 10 March 2010 (UTC)

(ec) *Keep as is. The term "map" is a synonym for the term "function". But "function" is the more common term, so for example, our main article on the topic is at function (mathematics), as opposed to map (mathematics). We currently have two articles which cover the topic of continuous functions an "elementary" version at continuous function, and an "advanced" version at continuous function (topology). It would be no more appropriate to move "Continuous function (topology)" to "continuous map", than it would be to move "continuous function" to "continuous map". Paul August ☎ 13:13, 10 March 2010 (UTC)

I wouldn't object to merging continuous function, and continuous function (topology) as suggested by Carl above. Paul August ☎ 14:14, 10 March 2010 (UTC)

• Comment. We also have the articles harmonic function and harmonic map that present much the same sort of strange distinction between "function" and "map", a distinction that I am not aware of existing outside Wikipedia. Regarding the current move, I vote to keep things where they are (although there is definitely a case to be made for sorting out harmonic function/map more satisfactorily). Sławomir Biały (talk) 14:06, 10 March 2010 (UTC)

Since there is no support for such a move, the current titles should be retained. To clarify what seems to me to be standard terminology, "function" is typically used for a scalar-valued morphism, whereas "map" is used when the target space is more general, e.g. a manifold. A map with values in R or C should certainly be called a function. Elements of homotopy groups of spheres are represented by maps, not functions. Tkuvho (talk) 14:11, 10 March 2010 (UTC)

For example, if you check the page homotopy group, you will not find any occurrence of the word "function" there, but plenty of maps. Tkuvho (talk) 14:21, 10 March 2010 (UTC)

I should clarify that perhaps "function" often connotes something scalar-valued and "map" a function with more general values, but this distinction is never one that I have seen actually enforced (outside of casual usage), and it is definitely a dangerous distinction on which to build a disambiguation. Having continuous function (topology) makes it very clear what the subject is, whereas continuous map introduces ambiguity. Sławomir Biały (talk) 14:26, 10 March 2010 (UTC)

Above you voted to "keep things the way they are". The disambiguation is already there, and we are not talking about creating a new one. If separate pages are kept, as you seem to suggest, the question becomes whether the second page should be "map", consistent with usage throughout topology, or "function (topology)", as is currently the case. Tkuvho (talk) 14:53, 10 March 2010 (UTC)

You are basing this on a connotative nuance, rather than a firm distinction (at least in my humble opinion): "Map" and "function" have exactly the same denotation. Do some authors favor the use of the term "map" in topology and other areas? Certainly, probably I do myself without even realizing it. Is this universal? No, probably not, and as I have said, I have never seen this as a hard terminological distinction. Then again, I am not an expert on what every mathematician has written about everything, but it seems to me that moving the well-disambiguated continuous function (topology) to continuous map increases rather than reduces the potential ambiguity. Sławomir Biały (talk) 15:06, 10 March 2010 (UTC)

I have no problem with that. It could be that my perception of a general preference for "map" in topology is more subjective than I (and the author of homotopy group) realized. Tkuvho (talk) 15:23, 10 March 2010 (UTC)

Note that we have a category "maps of manifolds" and a category "continuous mappings". People who contribute to technical pages in topology generally use "map". Tkuvho (talk) 16:25, 10 March 2010 (UTC)

I just checked a random page in those categories: Lefschetz fixed-point theorem. Not a word there about functions. Tkuvho (talk) 16:27, 10 March 2010 (UTC)

Sure; different fields often use differrent words for the same thing. I smiled when I opened my copy of Kelley's book and saw this:

The words "function", "map", "mapping", "correspondence", "operator", and "transformation" are synonymous.

There's a real-world example of having to disambiguate terms from different branches to place them all in the same general context, which predates Wikipedia by 45 years. — Carl (CBM · talk) 16:42, 10 March 2010 (UTC)

Tkuvho: did you have any objection to the merger? — Carl (CBM · talk) 16:45, 10 March 2010 (UTC)

No, not at all. "the cosine correspondence"? Tkuvho (talk) 16:51, 10 March 2010 (UTC)

Don't ask me. On a whim I searched for "continuous correspondence" on google books and it is actually used. — Carl (CBM · talk) 19:34, 10 March 2010 (UTC)

I have not seen the context of Kelley's comment, but I assume that when he says "synonymous", he does not necessarily mean "interchangeable". It may be useful to sort out when "function" and "map" are typically used in the literature, as well as in online encyclopedias, but I don't see too much support in this space for any changes. Tkuvho (talk) 08:40, 11 March 2010 (UTC)

Per the discussion above, I have merged this article with Continuous function and redirected it. The merged article is likely to need some work to make sure the content that was here fits appropirately within that article. — Martin (MSGJ · talk) 10:54, 30 March 2010 (UTC)