{"id":22,"date":"2009-02-13T10:29:47","date_gmt":"2009-02-13T14:29:47","guid":{"rendered":"http:\/\/blogs.williams.edu\/Morgan\/?p=22"},"modified":"2011-06-15T10:11:57","modified_gmt":"2011-06-15T15:11:57","slug":"topologies-on-r-of-all-possible-cardinalities","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2009\/02\/13\/topologies-on-r-of-all-possible-cardinalities\/","title":{"rendered":"Topologies on R of All Possible Cardinalities"},"content":{"rendered":"

In this guest column, David Thompson (Williams ’11) observes that there are topologies on R<\/strong> of all cardinalities from 2 to 2^{|R|}, assuming the Continuum Hypothesis, a result from the first topology tutorial session with his tutorial partner Andrew Lee and me.\u00a0In my topology tutorial, the 12 students meet weekly with me in groups of two or three to present and discuss the material.<\/p>\n

P.S. In the subsequent comment, fellow student Rob Silverstein generalizes part of the result to any space X.<\/p>\n

On the Cardinalities of Topologies of <\/em>R <\/strong>by David Thompson<\/span><\/em><\/p>\n

<\/em><\/p>\n

<\/span>In discussing the definition of a topology on a general set, a natural question arises regarding the sizes of allowable topologies.\u00a0 A topology is, after all, a collection<\/em> of subsets obeying certain properties, and the size of this collection can vary drastically. First recall the definition of a topology:<\/p>\n

A topology<\/em> T on a set X is a collection of open sets U such that:<\/p>\n

1.) X, \u00d8 \\in T,<\/p>\n

2.) T is closed under arbitrary unions\u00a0and finite intersections.<\/p>\n

Turning our attention to the real line, R<\/strong>, we see that the smallest possible topology is one containing the two members mandated by property 1; namely, T = {R<\/strong>, \u00d8}.\u00a0 We now claim that there is a topology on\u00a0R\u00a0containing three members.\u00a0 Let T = {R<\/strong>, \u00d8, {0}}.\u00a0 Looking at all possible unions and intersections of members of T, we see that T is indeed a topology.\u00a0 Similarly, we can generate a topology of any finite or countable cardinality by taking:<\/p>\n

T = {R<\/strong>, \u00d8, {0}, (-1,1), (-2,2),\u00a0 <\/sub>\u2026(-n,n)} for some n\\inN<\/strong>\u00a0(to achieve the same cardinality as N<\/strong> simply let n range through all of N<\/strong>).<\/p>\n

In fact, this method even provides us with a topology with the same cardinality as the continuum\u2014merely allow n to range through the positive reals instead of the positive integers.<\/p>\n

Finally, we introduce the notion of the discrete topology <\/strong>on a set X.\u00a0 The discrete topology<\/em> is the collection of all subsets of X.\u00a0 In the case of the real line, we have the collection of all subsets of R<\/strong>, which has cardinality\u00a02^{|R|}.\u00a0 Since the discrete topology is the largest topology we can have on R<\/strong>, it follows there is no topology with cardinality greater than\u00a02^{|R|}.\u00a0 What does this imply about the allowable sizes of topologies on R<\/strong>?\u00a0\u00a0Assuming the Continuum Hypothesis, which states that there is no set with cardinality strictly between that of N<\/strong>\u00a0and R<\/strong>, or R\u00a0<\/strong>and\u00a02^{|R|}, we see that we can designate a topology on R<\/strong>\u00a0of every size from 2 to\u00a02^{|R|}. \u00a0<\/p>\n

\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"

In this guest column, David Thompson (Williams ’11) observes that there are topologies on R of all cardinalities from 2 to , assuming the Continuum Hypothesis, a result from the first topology tutorial session with his tutorial partner Andrew Lee and me.<\/p>\n","protected":false},"author":269,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-22","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/22","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/269"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=22"}],"version-history":[{"count":1,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/22\/revisions"}],"predecessor-version":[{"id":650,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/22\/revisions\/650"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=22"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=22"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=22"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}