Transcription of Section 18. Continuous Functions
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Section 18. Continuous Functions
18. Continuous Functions1 Section isthefundamental concept in topology! When you hear that a coffee cup and a doughnut are topologically equivalent, this is really a claimabout the existence of a certain Continuous function (this idea is explored in depthin Chapter 12, Classification of Surfaces ). We start by reviewing some continuityideas from Analysis 1 (MATH 4217/5217). standard definition of continuity of a real valued function of a realvariable at a pointx0in the domain of the functionf,D(f), is as follows ( , page 2) a function andx0 D(f). Thenfiscontin-uous at pointx0iffor all >0 there exits ( )>0 such thatfor all|x x0|< ( ) andx D(f) we have|f(x) f(x0)|< .We then say thatfis Continuous a setA Riffis Continuous at each point following is a consequence of the previous definition (see Theorem 4-5in the Analysis 1 notes mentioned above):Theorem :X Y.
Jun 11, 2016 · In a graph, the structure is connectivity so that if vertices v and w are adjacent in G, then we require that π(v) and π(w) are adjacent in π(G) (and conversely). In a vector space the structure is linear combination, so we require av~1 + v~2 = w~, then aπ(v~1)+bπ(v~2) = w~. In a topological space, the structure is the collection of open sets.
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