## Dynamic topology |

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assume Axiom B-W set Cauchy sequence chain closed set closed subset cluster point collection compact metric space compact set compactification complement complete space component connected set connected subset connectedness Consider contradiction converges cut point define Definition dense diam diameter less directed family disjoint open sets distance function distinct points e-chain Euclidean Exercises exists a point exists an integer exists an open exponentially representable finite number finite subcover Fr(G given Hausdorff space Hence homeomorphism homotopic intersection Lemma lim G limit point locally compact locally connected continuum meets metric space monotone decreasing sequence noncut point nonempty normal open and closed open cover open set containing open set Ux perfectly separable space plane point inverses Proof quotient space region relative topology simple arc simple closed curve Suppose Theorem topological space Tr(p Trspace underfamily unicoherent uniformly continuous union upper semicontinuous