{"id":21050,"date":"2012-10-07T02:12:04","date_gmt":"2012-10-07T06:12:04","guid":{"rendered":"http:\/\/mathgradblog.williams.edu\/?p=21050"},"modified":"2013-04-05T01:41:20","modified_gmt":"2013-04-05T01:41:20","slug":"mission-counterexample","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2012\/10\/07\/mission-counterexample\/","title":{"rendered":"Mission Counterexample!"},"content":{"rendered":" An open ball inside the closed ball and isolated from its boundary.\n<p>Providing a counterexample can be the most challenging and frustrating exercise for me.\u00a0 For example, one of my recent homework exercises was to give an example of a metric space where the closure of an open ball, <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=B%28x_0%3Br%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"B(x_0;r)\" class=\"latex\" \/>, could differ from the closed ball, <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=bar%7BB%7D%28x_0%3Br%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"bar{B}(x_0;r)\" class=\"latex\" \/>.\u00a0 This is counter to naive intuition and experience: the closure of an open ball in <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=mathbb%7BR%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"mathbb{R}\" class=\"latex\" \/> is the closed ball.\u00a0 It was actually not difficult to find\u00a0a solution to the above exercise &#8211; can you find one?\u00a0 However, I am not content with finding one.\u00a0 If I can find one counterexample, then there is likely to be plenty of counterexamples.<\/p>\n<p>It is a source of pride for me if I can list two or three counterexamples when the exercise calls for only one.\u00a0 Usually the constraint of time limits me to just one counterexample.<!--more--><\/p>\n<p>Generally, I sketch what the counterexample must look like.\u00a0 For the above, the closure of the open ball is a proper subset of the closed ball.\u00a0 Therefore, the elements in the boundary of\u00a0 the closed ball must be isolated from the elements in the open ball.\u00a0 Sometimes, this approach is immediately fruitful; when it fails to be, it can be frustrating.\u00a0 With the glut of information available on the internet, it is very tempting to find a counterexample there.\u00a0 Normally, I give myself a week after I turn in my homework.\u00a0 If I can&#8217;t think of a counterexample by the end of the week,\u00a0 I&#8217;ll search for one on the internet.I&#8217;ll leave you with another exercise: find a sequence which converges to zero but is not in any <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=l%5Ep&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"l^p\" class=\"latex\" \/>-space for <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=1+le+p+%3C+infty&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"1 le p &lt; infty\" class=\"latex\" \/>.\u00a0 This is a little easier if you have the Cauchy condensation test in mind.\u00a0 I would like to hear some of your favorite counterexamples.<\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" ><\/div>","protected":false},"excerpt":{"rendered":"<p>Providing a counterexample can be the most challenging and frustrating exercise for me.\u00a0 For example, one of my recent homework exercises was to give an example of a metric space where the closure of an open ball, , could differ &hellip; <a href=\"https:\/\/blogs.ams.org\/mathgradblog\/2012\/10\/07\/mission-counterexample\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" data-url=https:\/\/blogs.ams.org\/mathgradblog\/2012\/10\/07\/mission-counterexample\/><\/div>\n","protected":false},"author":47,"featured_media":0,"comment_status":"open","ping_status":"closed","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":""},"categories":[8],"tags":[],"class_list":["post-21050","post","type-post","status-publish","format-standard","hentry","category-general"],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p3gbww-5tw","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/posts\/21050","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/users\/47"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/comments?post=21050"}],"version-history":[{"count":1,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/posts\/21050\/revisions"}],"predecessor-version":[{"id":22762,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/posts\/21050\/revisions\/22762"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/media?parent=21050"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/categories?post=21050"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/tags?post=21050"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}