totally bounded<\/a>, so for all n\u2208\u2115, we can cover M with finitely many open balls of radius 1\/n. The centers of all such balls are a countable set which we can call C. But C is dense, because an arbitrary point p\u2208M is a limit point of C: an \u03b5-neighborhood of p must contain the center of one the balls in our covering of M with \u03b5\/2-balls. Thus M contains a countable dense subset.<\/p>\n","protected":false},"excerpt":{"rendered":"It is now April! Did you know that April is one of the months in which every compact metric space is separable? Proof. Let it be April, and let M be a compact metric space. Because M is compact, it … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[7],"tags":[33],"_links":{"self":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/2221"}],"collection":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/comments?post=2221"}],"version-history":[{"count":1,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/2221\/revisions"}],"predecessor-version":[{"id":2222,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/2221\/revisions\/2222"}],"wp:attachment":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/media?parent=2221"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/categories?post=2221"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/tags?post=2221"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}