terms used in Royster notes

Terms mentioned in this binder's notes PhL 1.31.05 A. Metric Spaces Section metric, Taxi Cab metric, Discrete metric, sup metric norm Cauchy-Schwarz Inequality [~1830], Minkowski Inequality [1896] lub = sup and glb = inf subspace A of a metric space X diameter bounded distance between a point x and a subspace A product metric continuity , "Cauchy-Weierstrass continuity" open ball B(x1; r) N is a neighborhood of x1 open set, closed set family of all open sets is the topology for X generated by metric d Ac = X \ A = the complement of set A neighborhood definition of continuity limit of a sequence "true for almost all of S" "sequence definition of continuity" Properties of the open sets, Properties of the closed sets accumulation or limit point interior intA, boundary A, closure of S called S-bar B. Topological Spaces Section family, topology for X (X,F ) is called a topological space metric space appeared late, in 1906 (Maurice Frechet) In 1914 if was Hausdorff who coined the "topo space" discrete metric , singleton sets power set, usually written 2X = the discrete topology, the trivial topology finite complement topology = cofinite topology set A is closed if its complement X\A is open limit point A' is the derived set of all limit points dense , nowhere dense , separable basis of a topology. basic open sets, second countable local basis , first countable Every metric space is first countable homeomorphism topological property metrizable relative topology (aka subspace topology), relatively open sets, subspace hereditary property Hausdorff Space C. Compactness Section compact is a generalization of a set of the reals being both closed and bounded open cover , subcover Cantor's Nesting theorem locally compact at point a, locally compact Bolzano-Weierstrass Theorem Heine-Borel Theorem sequentially compact support, compact support