[理学]湖南师范大学 拓扑学top3.ppt

§26 Compact Spaces 1.Def.s (3) (0，1)is not compact, because {( ,1) | } is an open covering of R which has not finite subcovering. （5）X：any set , =the discrete topology, (X, ) is compact X is finite. Lemma Let Y be a subsp. of X. ThenY is compact if every covering of Y by sets open in X contains a finite subcollection covering . 2. properties Lemma If Y is a compact subspace of the Hausdorff space X and ，then open U Y, open s.t. . Cor. Let X be both compact and Hausdorff space. Then Example. R, =the finite complement top. (R, )is not Hausdorff. Theorem The product of finitely many compact sp.s is compact. Theorem Let X be a top. sp. Then X is compact a collection C of closed sets in X having the finite intersection property, . §27 Compact subspaces of the real Line 1、The compact sets of a simply ordered set with order topology Theorem A A is both closed and bounded in the Euclidean metric d or the square metric. 2. Extreme value Theorem 3. The uniform continuity Theorem §28 Limit point compactness 1. Limit Point Compactness 2. The relations between compactness and limit point compactness. 3 Sequentially compactness 4.Coincidences of the three types of compactness for metrizable sp.s. §29 Local compactness 1.Def. and Examples 2. Compactification 3. Equivalence condition of local compactness Chapter 4 Countability and Separation Axioms §30 The Countability Axioms 1．The First Countability Axiom 2.The 2nd countable Axiom 3. separable spaces §31 The separation Axioms 1. Def.s 2. Relations 3. properties Def. If a sp X has a countable basis for its topology, then X is said to satisfy the second countable axiom, or to be second-countable. Prop. Every second-countable sp is first-countable. Example 1 Rn is second-countable.{ } is a countable basis. Theorem Let X be first-countable/second-co