{"id":168,"date":"2006-09-27T20:51:48","date_gmt":"2006-09-27T20:51:48","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2006\/09\/27\/connectedness\/"},"modified":"2006-09-27T20:51:48","modified_gmt":"2006-09-27T20:51:48","slug":"connectedness","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2006\/09\/27\/connectedness\/","title":{"rendered":"Connectedness"},"content":{"rendered":"

Next stop on our tour of topology is the idea of *connectedness*. It’s an important concept that defines a lot of useful and interesting properties of topological spaces.
\nThe basic idea of connectedness is very simple and intuitive. If you think of a topology on a metric space like ℜ3<\/sup>, what connectedness means is, quite literally, connectedness in the physical sense: a space is connected if doesn’t consist of two or more pieces that never touch.
\nBeing more formal, there are several equivalent definitions:
\n* The most common one is the definition in terms of open and closed sets. It’s precise, concise, and formal; but it doesn’t have a huge amount of intuitive value. A topological space **T** is connected if\/f the only two sets in **T** that are both open *and* closed are **T** and ∅.
\n* The most intuitive one is the simplest set based definition: a topological space **T** connected if\/f **T** is *not* the union of two disjoint non-empty closed sets.
\n* One that’s clever, in that it’s got both formality and intuition: **T** is connected if the only sets in **T** with empty boundaries are **T** and ∅.
\nClosely related to the ida of connectedness is separation. A topological space is *separated* if\/f it’s not connected. (Profound, huh?)
\nSeparateness becomes important when we talk about *subspaces*, because it’s much easier to define when subspaces are *separated*; and they’re connected if they’re not separated.
\nIf A and B are subspaces of a topological space **T**, then they’re *separated in **T*** if and only if they are disjoint from each others closure. An important thing to understand here is that we are *not* saying that their *closures* are disjoint. We’re saying that A and B*<\/sup> are disjoint, and B and A*<\/sup> are disjoint, not that A*<\/sup> and B*<\/sup> are disjoint.
\nThe distinction is much clearer with an example. Let’s look at the topological space ℜ2<\/sup>. We can have *A* and *B* be *open* circles. Let’s say that *A* is the open circle centered on (-1,0) with radius one; so it’s every point whose distance from (-1,0) is *less than* 1. And let’s say that *B* is the open circle centered on (1,0), with radius 1. So the two sets are what you see in the image below. *A* is the shaded part of the green circle. The outline is the *boundary* of *A*, which is part of *A*<\/sup>*, but not part of *A* itself. *B* is the shaded part of the red circle; the outline is the boundary. Neither *A* nor *B* include the point (0,0). But both the *closure* of *A* and the *closure* of *B* contain (0,0). So *A* is disjoint from *B*<\/sup>*; and *B* is disjoint from *A*<\/sup>*. But *A*<\/sup>* is *not* disjoint from *B*<\/sup>*: they overlap at (0,0). *A* and *B* are separated, even though their closures overlap.
\n\"circles.jpg\"
\nThere’s also an even stronger notion of connectness for a topological space (or for subspaces): *path*-connectedness. A space **T** is *path connected* if\/f for any two points x,y ∈ **T**, there is a *continuous path* between x and y. Of course, we need to be a bit more formal than that; what’s a continuous path?
\nThere is a continuous path from x to y in **T** if there is a continuous function *f* from the closed interval [0,1] to **T**, where *f(0)=x*, and *f(1)=y*.<\/p>\n","protected":false},"excerpt":{"rendered":"

Next stop on our tour of topology is the idea of *connectedness*. It’s an important concept that defines a lot of useful and interesting properties of topological spaces. The basic idea of connectedness is very simple and intuitive. If you think of a topology on a metric space like ℜ3, what connectedness means is, quite […]<\/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-168","post","type-post","status-publish","format-standard","hentry","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\/168","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=168"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/168\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=168"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=168"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=168"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}