Due October 2 in class<\/p>\n
Section 20 problems 8a, 10
\nSection 21 problems 4, 6
\nSection 22 problems 2, 3, 4<\/p>\n
then<\/p>\n
A. Let \\(t\\) denote the taxicab metric on \\(\\mathbb{R}^n\\), given by \\(t((x_1,\\ldots, x_n),(y_1,\\ldots,y_n)) = \\sum_{i=1}^n \\left|x_i – y_i\\right| \\). Show that \\(t\\) and the euclidean metric induce the same topology on \\(\\mathbb{R}^n\\).<\/p>\n
B. Define \\(\\sim\\) on \\(\\mathbb{R}\\) by \\(x\\sim y\\) whenever \\(x-y \\in\\mathbb{Z}\\). The quotient \\(\\mathbb{R}\/\\sim\\)is also denoted as \\(\\mathbb{R}\/\\mathbb{Z}\\). Does \\(\\mathbb{R}\/\\mathbb{Z}\\) look like any topological space that we are familiar with? Justify your answer. Secondly, define a partition \\(\\mathbb{R}^\\ast\\) consisting of the singletons \\(\\{x\\}\\) if \\(x\\not\\in \\mathbb{Z}\\) and the set \\(\\mathbb{Z}\\). Do you think the quotient space \\(\\mathbb{R}^\\ast\\) is homeomorphic to \\(\\mathbb{R}\/\\mathbb{Z}\\) (Do your best to answer this last question…it will be difficult to prove this rigorously. Appeal to your intuition here)? <\/p>\n
C. Recall from class the line with two origins. It inherits a quotient topology from \\(\\mathbb{R}\\times 1 \\sqcup \\mathbb{R}\\times -1 \\subset \\mathbb{R}^2\\), and it inherits the subspace topology from \\(\\mathbb{R}^2\\). Compare and contrast the two topologies on the line with two origins.<\/p>\n
D. Is a quotient of a Hausdorff space necessarily Hausdorff? Prove it or provide a counterexample.<\/p>\n","protected":false},"excerpt":{"rendered":"
Due October 2 in class Section 20 problems 8a, 10 Section 21 problems 4, 6 Section 22 problems 2, 3, 4 then A. Let \\(t\\) denote the taxicab metric on \\(\\mathbb{R}^n\\), given by \\(t((x_1,\\ldots, x_n),(y_1,\\ldots,y_n)) = \\sum_{i=1}^n \\left|x_i – y_i\\right| \\). Show that \\(t\\) and the euclidean metric induce the same topology on \\(\\mathbb{R}^n\\). B. […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[2],"tags":[],"_links":{"self":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2013mat661\/wp-json\/wp\/v2\/posts\/56"}],"collection":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2013mat661\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2013mat661\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2013mat661\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2013mat661\/wp-json\/wp\/v2\/comments?post=56"}],"version-history":[{"count":16,"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2013mat661\/wp-json\/wp\/v2\/posts\/56\/revisions"}],"predecessor-version":[{"id":72,"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2013mat661\/wp-json\/wp\/v2\/posts\/56\/revisions\/72"}],"wp:attachment":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2013mat661\/wp-json\/wp\/v2\/media?parent=56"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2013mat661\/wp-json\/wp\/v2\/categories?post=56"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2013mat661\/wp-json\/wp\/v2\/tags?post=56"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}