When we talk about topology, in general, the way we talk about it is in terms of *shapes*: geometric objects and spaces, surfaces, bodies that enclose things, etc. We talk about the topology of a *torus*, or a *coffee mug*, or a *sphere*.
\nBut the topology we’ve talked about so far doesn’t talk about shapes or surfaces. It talks about open sets and closed sets, about neighborhoods, even about filters; but we haven’t touched on how this relates to our *intuitive* notion of shape.
\nToday, we’ll make a start on the idea of surface and shape by defining what *interior* and *boundary* mean in a topological space.<\/p>\n
When we talk about topology, in general, the way we talk about it is in terms of *shapes*: geometric objects and spaces, surfaces, bodies that enclose things, etc. We talk about the topology of a *torus*, or a *coffee mug*, or a *sphere*. But the topology we’ve talked about so far doesn’t talk about shapes […]<\/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-149","post","type-post","status-publish","format-standard","hentry","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-2p","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/149","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=149"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/149\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=149"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=149"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=149"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nWe need to start with some concepts based on a notion of distance. In a topological space, the basic notion of distance is built on neighborhoods. For each concept, I’ll start by describing it informally in terms of metric spaces, because that’s where our intuitions of distance come from.
\nFirst, we need to be able to talk about what it means to be *arbitrarily close* to a set. In a metric space with metric function d, we can define the distance between a point and a set *S* by a function dS<\/sub>(x) = min({d(x,p) | p ∈ *S*}). So the distance from a point x to a set *S* is the distance between *x* and the *closest* member of *S*.
\nWith that, we can talk about being *arbitrarily close* to a set. In a metric space, a point *p* is *arbitrarily close* to a set *S* if ∀ x ∈ ℜ > 0, dS<\/sub>(p) 0″. I was so busy focusing on the intuition that I neglected to make the formal part precise.)*
\nThe set of all points that are arbitrarily close to a set *S* in a metric space are called the *closure* of S. That’s usually written by *S* with a horizontal line over it by topologists; but since I can’t find any good way to use that notation in HTML, I’ll use the typical computer science notation for closure, and write *S*<\/sup>*. Intuitively, if *S* is an open set, then *S*<\/sup>* is the *closed* set containing *S* and its *boundary*.
\nWhat’s the corresponding notion in an arbitrary (i.e., not necessarily metric) topological space? Suppose we’ve got a topological space (**T**, τ); and a subset *S* of **T**. The closure *S*<\/sup>* of *S* is the set of all points p where for every neighborhood of p, N(p), N(p) ∩ *S* ≠ ∅: that is, all points where one of their neighborhoods contains at least one point in *S*. It’s the same basic idea as the metric space closure, but it’s based on neighborhoods instance of distance metrics. In any metric space that’s also a topological space, the topological and metric closures are the same. In fact, we can even formalize the idea that I mentioned up above about closed sets: a set *S* is a closed set if, and only if *S*<\/sup> = S*. The closure of *S* is the *smallest* closed set containing *S*. *(Note: originally, this paragraph contained a typo: I used “∪” instead of “∩”; In HTML, that’s “cup” instead of “cap”.)*
\nIt’s a pretty easy to prove set of theorems that there are certain properties of a topological space involving closures. Given a topological space is (**T**, τ):
\n1. ∅*<\/sup> = ∅. *(The closure of the empty set is the empty set.)*
\n2. **T***<\/sup> = **T**. *(The closure of the set all points in a topological space is the set of all points in the topological space).*
\n3. ∀ S ⊂ **T**: S ⊆ S*<\/sup>. *(Every subset of **T** is a subset of its closure.)*
\n4. ∀ A,B ⊂ **T**: (A ∪ B)*<\/sup> = A*<\/sup>∪B*<\/sup>. *(The closure of a union of two subsets of **T** is union of their closures; closure in invariant over set union.)*
\n5. ∀ S ⊂ **T**: S*<\/sup> = S**<\/sup><\/sup>.
\n*(Taking the closure of a closure doesn’t do anything.)*
\nNow that we can talk about the closure of sets in a topological space, we can move on to the definitions we really care about: interior and boundary.
\nIn a *metric* space, we know what the interior is, intuitively. We can define any shape in a metric space by the combination of unions and intersections of *open balls*; so we can define interior for open-balls, and go from there. If B(p,x) is an open ball of size x around a point p, then a point z is in the *interior* of B(p,x) if d(z,x) < p. The extension of this over unions and intersections works exactly as your intuition would predict; the interior of a set in a metric space matches your intuition of *inside*.
\nWhat this means is that the interior of a set is a kind of *opposite* of the closure: the closure of a set S was the *smallest* closed set that includes S; the *interior* of a set S is the *largest* open set that is included by S.
\nAnd that's exactly the definition that we can if we expand to topological spaces: in a topological space (**T**, τ), a point p is in the interior of a set *S*⊂**T** if and only if *S* is a neighborhood of p. Work out the set of points p for which *S* is a neighborhood, and it's the *largest* open set included by *S*:
\nGiven a subset *S* of a topological space (**T**,τ), the *interior* of *S* is the union of all open sets contained in *S*. The interior of the set S is often written Int(S).
\n*(Note: the original version of the above three paragraphs contained errors in the phrasing; they originally said "the interior of a set S is the largest open set that includes S, where it should have been the largest open set that *is included by* S.)*
\nAnd last but definitely not least; the *boundary* of a set *S* in a topological space (**T**,τ) is the intersection of the *closure* of *S* and the closure of the complement of S:
\nBoundary(S) = S*<\/sup> ∩ (S-1<\/sup>)*<\/sup>
\nOnce again, if we were to look at this in a metric space, it’s *exactly* what our intuition would call the boundary or surface of S. The surface of a sphere is the set of points which is arbitrarily close to both the *exterior* of the sphere and the interior of the sphere. It’s the set of points that forms a *boundary* between the interior and the exterior.<\/p>\n","protected":false},"excerpt":{"rendered":"