{"id":296,"date":"2016-09-29T13:19:49","date_gmt":"2016-09-29T17:19:49","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=296"},"modified":"2017-08-22T17:01:36","modified_gmt":"2017-08-22T21:01:36","slug":"are-all-compact-sets-closed-and-bounded","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2016\/09\/29\/are-all-compact-sets-closed-and-bounded\/","title":{"rendered":"Are all compact sets closed and bounded?"},"content":{"rendered":"<p>As shown in the textbook all compact sets in the space of real numbers must be closed and bounded (Abbot 96). However, topological spaces go far beyond just a set that is a subset of the reals. This brings to our attention the idea that the proof for sets being compact if and only if they are closed and bounded may not hold true for other topological spaces. So an obvious question presents itself:<\/p>\n<p style=\"text-align: center;\"><strong><em>Are all compact sets closed and bounded?<\/em><\/strong><\/p>\n<p>To fully understand and comprehend this question we need a full understanding of what it means to be <em>compact<\/em>, <em>closed<\/em>, and<em> bounded<\/em>.<\/p>\n<p><strong>Compact:\u00a0<\/strong><\/p>\n<ul>\n<li>First, for a set to be compact, every sequence in the set must have a subsequence that converges to a limit that is also contained in the set.<\/li>\n<li>Further, if a set is a compact set it means that given any open cover we can find a finite subcover that covers the\u00a0set (Abbot 96).<\/li>\n<\/ul>\n<p><strong>Bounded: <\/strong>A set is bounded if there is a bound that the all values in the set have an absolute value less than or equal to that bound.<\/p>\n<p><strong>Closed:<\/strong><\/p>\n<ul>\n<li>First, a set is closed if it contains all of its limit points.<\/li>\n<li>Further, a set\u00a0is closed if and only if the set&#8217;s complement is open (Abbot 90).<\/li>\n<\/ul>\n<p>With everything defined it is now reasonable to address answers to the question.<\/p>\n<p><strong>Claim:\u00a0<\/strong>There exist\u00a0compact sets which are not\u00a0closed.<\/p>\n<p>The easiest way to demonstrate this will be to find a topological space within which there\u00a0is a compact set that is not closed. Take the general set <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=X+%3D+%5C%7Ba%2Cb%5C%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"X = &#92;{a,b&#92;}\" class=\"latex\" \/> and\u00a0a potential\u00a0topology on\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=X%2C+%5C+%5Ctau+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"X, &#92; &#92;tau \" class=\"latex\" \/> = <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5C%7B%5Cemptyset%2C+%5C%7Ba%5C%7D%2C+%5C%7Ba.b%5C%7D%5C%7D+%C2%A0&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;{&#92;emptyset, &#92;{a&#92;}, &#92;{a.b&#92;}&#92;} \" class=\"latex\" \/>. First, we need to show that\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5C%7B%5Cemptyset%2C+%5C%7Ba%5C%7D%2C+%5C%7Ba.b%5C%7D%5C%7D+%C2%A0&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;{&#92;emptyset, &#92;{a&#92;}, &#92;{a.b&#92;}&#92;} \" class=\"latex\" \/> is actually a topology. To be a topology it needs to satisfy\u00a0three conditions from the definition (which can be found\u00a0on <a href=\"http:\/\/mathworld.wolfram.com\/TopologicalSpace.html\">mathworld.wolfram.com<\/a>.\u00a0First, the empty set and the set itself must be in the topology which holds true here because <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cemptyset+%5Cin+%5Ctau+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;emptyset &#92;in &#92;tau \" class=\"latex\" \/> and <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5C%7Ba%2Cb%5C%7D+%5Cin+%5Ctau+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;{a,b&#92;} &#92;in &#92;tau \" class=\"latex\" \/>. Next, the union of\u00a0any combination of\u00a0sets in the topology must be in the topology which is true because <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cemptyset+%5Ccup+%5C%7Ba%5C%7D+%5Ccup+%5C%7Ba%2Cb%5C%7D+%3D+%5C%7Ba%2Cb%5C%7D+%5Cin+%5Ctau+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;emptyset &#92;cup &#92;{a&#92;} &#92;cup &#92;{a,b&#92;} = &#92;{a,b&#92;} &#92;in &#92;tau \" class=\"latex\" \/>,\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cemptyset+%5Ccup+%5C%7Ba%5C%7D+%3D+%5C%7Ba%5C%7D+%5Cin+%5Ctau+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;emptyset &#92;cup &#92;{a&#92;} = &#92;{a&#92;} &#92;in &#92;tau \" class=\"latex\" \/>,\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cemptyset+%5Ccup+%5C%7Ba%2Cb%5C%7D+%3D+%5C%7Ba%2Cb%5C%7D+%5Cin+%5Ctau+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;emptyset &#92;cup &#92;{a,b&#92;} = &#92;{a,b&#92;} &#92;in &#92;tau \" class=\"latex\" \/>, and\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5C%7Ba%5C%7D+%5Ccup+%5C%7Ba%2Cb%5C%7D+%3D+%5C%7Ba%2Cb%5C%7D+%5Cin+%5Ctau+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;{a&#92;} &#92;cup &#92;{a,b&#92;} = &#92;{a,b&#92;} &#92;in &#92;tau \" class=\"latex\" \/> so all of the possible combinations of the unions of the topology are in the topology. Finally, the topology must contain the intersection of\u00a0any finite combination\u00a0of the sets in the topology which is again true because\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cemptyset+%5Ccap+%5C%7Ba%5C%7D+%5Ccap+%5C%7Ba%2Cb%5C%7D+%3D+%5Cemptyset+%5Cin+%5Ctau+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;emptyset &#92;cap &#92;{a&#92;} &#92;cap &#92;{a,b&#92;} = &#92;emptyset &#92;in &#92;tau \" class=\"latex\" \/>, <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cemptyset+%5Ccap+%5C%7Ba%5C%7D+%3D+%5Cemptyset+%5Cin+%5Ctau+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;emptyset &#92;cap &#92;{a&#92;} = &#92;emptyset &#92;in &#92;tau \" class=\"latex\" \/>, <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cemptyset+%5Ccap+%5C%7Ba%2Cb%5C%7D+%3D+%5Cemptyset%C2%A0%5Cin+%5Ctau+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;emptyset &#92;cap &#92;{a,b&#92;} = &#92;emptyset\u00a0&#92;in &#92;tau \" class=\"latex\" \/>, and <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5C%7Ba%5C%7D+%5Ccap+%5C%7Ba%2Cb%5C%7D+%3D+%5C%7Ba%5C%7D+%5Cin+%5Ctau+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;{a&#92;} &#92;cap &#92;{a,b&#92;} = &#92;{a&#92;} &#92;in &#92;tau \" class=\"latex\" \/>\u00a0so all possible combinations of intersections are in the topology. Since these three conditions are met it\u00a0follows that <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Ctau+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;tau \" class=\"latex\" \/> is in fact a topology.<\/p>\n<p>The next step is to show that there is a set in the topology that is compact but not closed. Choose the set <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5C%7Ba%5C%7D+%5Cin+%5Ctau+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;{a&#92;} &#92;in &#92;tau \" class=\"latex\" \/>. This set is compact because there is only one sub-sequence in {a} which obviously must converge to a so every sequence has a sub-sequence that converges to a limit inside the set,\u00a0hence it is compact. However, the set is not closed because the complement of all closed sets\u00a0must be\u00a0open, and\u00a0the complement <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5C%7Ba%5C%7D%5Ec+%3D+%5C%7Bb%5C%7D+%5Cnotin+%5Ctau&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;{a&#92;}^c = &#92;{b&#92;} &#92;notin &#92;tau\" class=\"latex\" \/>\u00a0is not in the topology, hence not open. Since, <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%7Ba%7D%5Ec&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"{a}^c\" class=\"latex\" \/>\u00a0is not open, it follows that\u00a0{a} is not closed.\u00a0So this set and topology provide a counterexample that not all compact sets need to be closed.<\/p>\n<p>The set I used to prove that a compact set does not have to be closed is known as a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Sierpi%C5%84ski_space\">Sierpinski Space<\/a>. The Sierpinski Space is a finite topological space with two points, only one of which is closed, which is key in proving that not all compact sets are closed outside of the reals. However, this simple topological space is crucial for a lot of general topology. For example, it is the smallest topological space that is neither trivial or discrete that can be constructed. Trivial means that the space contains more than just the empty set and the whole\u00a0space while discrete means that it is a space that forms a discontinuous sequence. Also, it contains qualities that often contradict standard properties of sets in <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BR%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;mathbb{R}\" class=\"latex\" \/>, such as being compact and not closed but also in more advanced topology. With its very many unique qualities it is an extremely useful tool in proving theorems for general topology. While most of the uses of this space are advanced it is interesting to note that what seems to be a simple space is an extremely powerful tool for mathematics.<\/p>\n<p><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/images.tutorvista.com\/cms\/images\/131\/topological-space.png?w=600\" alt=\"Topological Space\" \/><\/p>\n<p>Now to address boundedness look at the diagram above. To demonstrate that a compact set need not be bounded focus on the two outer most circles, the topological space and metric spaces. A metric space is a space where the distance between any two\u00a0elements is defined (<a href=\"http:\/\/www-history.mcs.st-and.ac.uk\/~john\/MT4522\/Lectures\/L5.html\">more details here<\/a>). <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BR%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;mathbb{R}\" class=\"latex\" \/> is a metric space with distance between members being |a-b| where <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=a%2Cb+%5Cin+%5Cmathbb%7BR%7D+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"a,b &#92;in &#92;mathbb{R} \" class=\"latex\" \/>. As you can see in the diagram\u00a0above not all topological spaces are metric spaces. This means that there are topological spaces where the distance between two members of the set are undefined. As mentioned earlier a set is bounded if there is a bound that the all values in the set have an absolute value less than or equal to that bound which means <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cexists+M+%3E+0+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;exists M &gt; 0 \" class=\"latex\" \/> such that the distance between elements <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=d%28x%2Cy%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"d(x,y)\" class=\"latex\" \/> is less than M, <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=d%28x%2Cy%29+%5Cleq+M+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"d(x,y) &#92;leq M \" class=\"latex\" \/>\u00a0when\u00a0x and y are elements of the set in question. However, as stated before the distance between members of a topological space that is not metricized is undefined so you cannot say that <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=d%28x%2Cy%29+%5Cleq+M+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"d(x,y) &#92;leq M \" class=\"latex\" \/> because d(x,y) is undefined and therefore you cannot say that any set in the space is bounded. Since, <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BR%7D+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;mathbb{R} \" class=\"latex\" \/> is a metric spaces and we are working with topological spaces that are not metric Theorem 3.3.4 (Characterization of Compactness in <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BR%7D+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;mathbb{R} \" class=\"latex\" \/>) no longer applies. So a compact set does not need to be closed and bounded.<\/p>\n<p><strong>Claim:\u00a0<\/strong>There exist\u00a0compact sets which are not\u00a0bounded.<\/p>\n<p>One\u00a0example of a topological space within which there are compact, unbounded sets,\u00a0is the <a href=\"http:\/\/mathworld.wolfram.com\/ZariskiTopology.html\">Zariski Topology<\/a>. A Zariski topology \u201cis a topology that is well-suited for the study of polynomial equations in algebraic geometry, since a Zariski topology has many fewer open sets than in the usual metric topology. In fact, the only closed sets are the algebraic sets, which are the zeros of polynomials.\u201d<\/p>\n<p>A slightly more understandable\u00a0example is\u00a0the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Cofiniteness\">cofinite topology<\/a> on <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BR%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;mathbb{R}\" class=\"latex\" \/>, wherein open sets are defined\u00a0as those subsets of <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BR%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;mathbb{R}\" class=\"latex\" \/>\u00a0that\u00a0can be written as <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=A+%3D+%5Cmathbb%7BR%7D+%5Csetminus+%5C%7Bx_1%2C+x_2%2C...%2C+x_n+%5C%7D+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"A = &#92;mathbb{R} &#92;setminus &#92;{x_1, x_2,..., x_n &#92;} \" class=\"latex\" \/> for some finite <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=n+%5Cin+%5Cmathbb%7BN%7D+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"n &#92;in &#92;mathbb{N} \" class=\"latex\" \/>. Recalling the definition of compactness,\u00a0suppose\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5C%7BO_1%2C+O_2%2C...%2C%5C%7D+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;{O_1, O_2,...,&#92;} \" class=\"latex\" \/> is an open cover\u00a0for\u00a0some set <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=C+%5Csubseteq+%5Cmathbb%7BR%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"C &#92;subseteq &#92;mathbb{R}\" class=\"latex\" \/>. Then the first set <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=O_1+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"O_1 \" class=\"latex\" \/> contains all but at most finitely many points of the set C.\u00a0Then we can choose a set <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=O_n+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"O_n \" class=\"latex\" \/> from the original covering that contains <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=x_1&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"x_1\" class=\"latex\" \/>, the first of these\u00a0points missed by <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=O_1&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"O_1\" class=\"latex\" \/>.\u00a0Then another set <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=O_m+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"O_m \" class=\"latex\" \/> must cover\u00a0the second of these points <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=x_2&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"x_2\" class=\"latex\" \/>, etcetera.\u00a0Continuing\u00a0this process\u00a0for all the points missed by <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=O_1+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"O_1 \" class=\"latex\" \/> and you will get a finite\u00a0subcover of the original open cover. Hence the set is compact. Notice that no restriction was placed on the set <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=C+%5Csubseteq+%5Cmathbb%7BR%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"C &#92;subseteq &#92;mathbb{R}\" class=\"latex\" \/>, so we have actually just shown that <strong>EVERY<\/strong> subset is compact in this cofinite topology. Hence, there is are unbounded\u00a0sets C which\u00a0are compact.<\/p>\n<p>In class and in chapter 3 we deal entirely with sets on the reals. However, as shown by the example above in more general terms the same rules do not apply. As shown in the reals one of the defining factors of a compact set is that it must be closed and bounded.\u00a0The fact that\u00a0a compact set is closed and bounded in the reals is known as the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Heine%E2%80%93Borel_theorem\">Heine-Borel theorem<\/a>. The fact that there is a very well-known and important theorem that only holds true for the reals demonstrates that studying properties in the reals is not only unique but also very important to the field of mathematics which is why there is a whole course called Real Analysis.<\/p>\n<p>Sources:<\/p>\n<p>http:\/\/math.stackexchange.com\/questions\/239998\/compact-sets-are-closed<\/p>\n<p>https:\/\/en.wikipedia.org\/wiki\/Sierpi%C5%84ski_space<\/p>\n<p>Abbott, Stephen. &#8220;Basic Topology of \u211d.&#8221; Understanding Analysis. New York: Springer, 2015. 85-109. Print.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>As shown in the textbook all compact sets in the space of real numbers must be closed and bounded (Abbot 96). However, topological spaces go far beyond just a set that is a subset of the reals. This brings to our attention the idea that the proof for sets being compact if and only if [&hellip;]<\/p>\n","protected":false},"author":3014,"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":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[44423],"tags":[],"class_list":["post-296","post","type-post","status-publish","format-standard","hentry","category-fall-2016"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-4M","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/296","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/users\/3014"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=296"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/296\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=296"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=296"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=296"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}