Topology by Marco Manetti

Author:Marco Manetti
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

7.23

(, ) Let be a locally finite open cover of a normal space . Prove that there’s an open cover such that for every . (Hint: use Zorn’s lemma on the set of pairs , where , is open with for every , and .)

7.24

() Use Exercise 6.​28 and the arguments in Exercise 3.​62 to show that the lower-limit plane is not normal, and hence neither paracompact.

7.6 Separation Axioms

Being Hausdorff or normal are perhaps the most familiar instances of what are called in the literature separation axioms. The imagination of mathematicians has produced many more properties of this kind, and the next definition summarises only the most renowned, which Tietze introduced in 1921 under the name Trennbarkeitsaxiome.

Definition 7.34

A space is called: T0 if distinct points have distinct closures;

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