{"id":142,"date":"2006-09-04T16:22:22","date_gmt":"2006-09-04T16:22:22","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2006\/09\/04\/back-to-topology-continuity-corrected\/"},"modified":"2006-09-04T16:22:22","modified_gmt":"2006-09-04T16:22:22","slug":"back-to-topology-continuity-corrected","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2006\/09\/04\/back-to-topology-continuity-corrected\/","title":{"rendered":"Back to Topology: Continuity (CORRECTED)"},"content":{"rendered":"

*(Note: in the original version of this, I made an absolutely **huge** error. One of my faults in discussing topology is scrambling when to use forward functions, and when to use inverse functions. Continuity is dependent on properties defined in terms of the *inverse* of the function; I originally wrote it in the other direction. Thanks to commenter Dave Glasser for pointing out my error. I’ll try to be more careful in the future!)*
\nSince I’m back, it’s time to get back to topology!
\nI’m going to spend a bit more time talking about what continuity means; it’s a really important concept in topology, and I don’t think I did a particularly good job at explaining it in my first attempt.
\nContinuity is a concept of a certain kind of *smoothness*. In non-topological mathematics, we define continuity with a very straightforward algebraic idea of smoothness. A standard intuitive definition of a *continuous function* in algebra is “a function whose graph can be drawn without lifting your pencil”. The topological idea of continuity is very much the same kind of thing – but since a topological space is just a set with some additional structure, the definition of continuity has to be generalized to the structure of topologies.
\nThe closest we can get to the algebraic intuition is to talk about *neighborhoods*. We’ll define them more precisely in a moment, but first we’ll just talk intuitively. Neighborhoods only exist in topological metric spaces, since they end up being defined in terms of distance; but they’ll give us the intuition that we can build on.
\nLet’s look at two topological spaces, **S** and **T**, and a function f : **S** → **T** (that is, a function from *points* in **S** to *points* in **T**). What does it mean for f to be continuous? What does *smoothness* mean in this context?
\nSuppose we’ve got a point, *s* ∈ **S**. Then f(*s*) ∈ **T**. If f is continuous, then for any point p in **T** *close to f(s)*, f-1<\/sup>(p) will be *close to* *s*. What does close to mean? Pick any distance – any *neighborhood* N(f(s)) in **T** – no matter how small; there will be a corresponding neighborhood of M(*s*) around s in **S** so that for all points p in N(f(s)), f-1<\/sup> will be in M(*s*). If that’s a bit hard to follow, a diagram might help:
\n\"continuity.jpg\"
\nTo be a bit more precise: let’s define a neighborhood. A neighborhood N(p) of a point p is a set of points that are *close to* p. We’ll leave the precise definition of *close to* open, but you can think of it as being within a real-number distance in a metric space. (*close to* for the sake of continuity is definable for any topological space, but it can be a strange concept of close to.)
\nThe function f is continuous if and only if for all points f(s) ∈ **T**, for all neighborhoods N(f(s)) of f(s), there is some neighborhood M(s) in **S** so that f(M(s)) ⊆ N(f(s)). Note that this is for *all* neighborhoods of *all* points in **T** mapped to by f – so no matter how small you shrink the neighborhood around f(s), the property holds – and it implies that as the neighborhood in **T** shrinks, so does the corresponding neighborhood in **S**, until you reach the single points f(s) and s.
\nWhy does this imply *smoothness*? It means that you can’t find a set of points in the range of f in **T** that are close together, but that weren’t close together in **S** before being mapped by f. f won’t put things together that weren’t together originally. And it won’t pull things apart that weren’t
\nclose together originally. *(This paragraph was corrected to be more clear based on comments from Daniel Martin.)*
\nFor a neat exercise: go back to the category theory articles, where we defined *initial* and *final* objects in a category. There are corresponding notions of *initial* and *final* topologies in a topological space for a set. The definitions are basically the same as in category theory – the arrows from the initial object are the *continuous functions* from the topological space, etc.<\/p>\n","protected":false},"excerpt":{"rendered":"

*(Note: in the original version of this, I made an absolutely **huge** error. One of my faults in discussing topology is scrambling when to use forward functions, and when to use inverse functions. Continuity is dependent on properties defined in terms of the *inverse* of the function; I originally wrote it in the other direction. […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[24,65],"tags":[],"class_list":["post-142","post","type-post","status-publish","format-standard","hentry","category-goodmath","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-2i","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/142","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=142"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/142\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=142"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=142"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=142"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}