The topology posts have been extremely abstract lately, and from some of the questions
\nI’ve received, I think it’s a good idea to take a moment and step back, to recall just
\nwhat we’re talking about. In particular, I keep saying “a topological space is just a set
\nwith some structure” in one form or another, but I don’t think I’ve adequately maintained
\nthe *intuition* of what that means. The goal of today’s post is to try to bring back
\nat least some of the intuition.<\/p>\n
The topology posts have been extremely abstract lately, and from some of the questions I’ve received, I think it’s a good idea to take a moment and step back, to recall just what we’re talking about. In particular, I keep saying “a topological space is just a set with some structure” in one form or […]<\/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":[65],"tags":[],"class_list":["post-251","post","type-post","status-publish","format-standard","hentry","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-43","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/251","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=251"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/251\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=251"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=251"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=251"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nSo let’s recall just what a topological space is. Our definition from the [very beginning of
\nthe topology series was:][top-space]
\nA topological space is a set **X**, and a collection **T** of subsets of **X** where the following conditions hold:
\n1. ∅ ∈ **T** and **X** ∈ **T** *(The empty set and the entire set **X** are both members of **T**.)
\n2. (∀ C ∈ **2****T**<\/sup>): (∀ c ∈ C : ∪(c ∈ C)<\/sub>c ∈ **T**) *(The union of any set of elements of **T** must be a set in **T**.)*
\n3. ∀ s,t ∈ **T**: s ∩ t ∈ **T** *(The intersection of any pair of members of **T** is also a member of **T**.)*
\n**T** is the structure imposed on the set **X** that we’ve been talking about. But just what does that mean? It’s really a very fancy way of taking the concept of *closeness* or *adjacency* and abstracting it out so the concept of *distance* isn’t needed. In a topological space, we don’t care whether we can measure *how far* it is from a point *A* to a point *B*; but we *do* care
\nwhether we can meaningfully ask “Is B closer to A than C?” or “Is A adjacent to B?”. The
\nstructure of the open subsets in a topological space gives us a way of talking about that.
\nHow can we answer those questions? By playing with neighborhoods – that is, expanding “shells” of points around a particular given point. Suppose we want to ask “**Is *B* closer to *A* than it is to *C*?**”
\nTake a sequence *S* of expanding subsets around *B* something like the open balls in a metric space – that is, a sequence of subsets that are uniformly growing larger, but always including all of the points in the subsets that precede them. If *A* becomes an element of the sets in the sequence *before* *C* does, then *with respect to* the sequence *S*, *A* is closer to *B* than *C* is. In a topological space, you *cannot* in general define something like *closer to* in a universal way; there are many ways that the open sets can be constructed, and it’s entirely possible to have *many different* ways of describing closeness based on different
\nconstructions, and there’s no reason to prefer one of them over the other.
\nThat basic concept – what points are *next to* what points, and what points are *close to* what other points – is what’s defined by the open-set structure of the topological space. The notion of “close to” is based completely on subset inclusion relationships; you can’t necessarily assign a number to the distance between two points (if you could, we’d call it a metric space!), but you can always look at the subset inclusion relationships to understand where the points lie in relation to each other.
\nTo bring this forward a bit, in my messed up post about sheaves, one of the key ideas was
\nthe gluing axiom. The gluing axiom says, very basically, that you can map between *sets* in an overlap between two sections; it does *not* do a coordinate transformation. That misunderstanding was caused by a combination of some genuinely subtle distinctions, and some
\ndreadful sloppiness on my part.
\nWhen we talked about gluing manifolds, what we were doing is forming manifolds by mapping
\nsections of *euclidean spaces* onto manifolds, and gluing them together with coordinate
\ntransformations. That *is* gluing, and the theoretical basis for it *is* sheaves and
\nthe gluing axiom that allows them to be combined. But the important distinction is that what we were gluing was sections of euclidean spaces – and euclidean spaces have a standard metric, and we describe the glue maps in terms of that standard metric.
\n[top-space]: http:\/\/scienceblogs.com\/goodmath\/2006\/08\/topological_spaces.php<\/p>\n","protected":false},"excerpt":{"rendered":"