By R. H. Bing, Ralph J. Bean

ISBN-10: 0691080569

ISBN-13: 9780691080567

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Prove! 3 Net Convergence and Closure Our claim is that nets ‘fully describe’ the structure of a topological space. 4 Given (X, T ), A ⊆ X, p ∈ X: p ∈ A¯ iff there exists a net in A converging to p. 30 Proof If some net of points of A converges to p, then every neighbourhood ¯ of p contains points of A (namely, values of the net) and so we get p ∈ A. Conversely, if p is a closure point of A then, for each neighbourhood N of p, it will be possible to choose an element aN of A that belongs also to N .

Given closed K ⊆ X1 , then K is compact whence f (K) is compact and so f (K) is closed. Thus f is a closed map. 6 (X, T ) is T2 iff no net in X has more than one limit. Proof (i) ⇒ (ii): Let x = y in X; by hypothesis, there exist disjoint neighbourhoods U of x, V of y. Since a net cannot eventually belong to each of two disjoint sets, it is clear that no net in X can converge to both x and y. (ii) ⇒ (i): Suppose that (X, T ) is not Hausdorff and that x = y are points in X for which every neighbourhood of x intersects every neighbourhood of y.

By the previous remark, we can now construct: (i) G 1 ∈ T : G¯0 ⊆ G 1 , G¯1 ⊆ G1 . 2 2 2 (ii) G 1 , G 3 ∈ T : G¯0 ⊆ G 1 , G¯1 ⊆ G 1 , G¯1 ⊆ G 3 , G¯3 ⊆ G1 . 4 4 4 4 2 2 4 4 (iii) . . and so on! Thus we get an indexed family of open sets {Gr : r = m , 0 ≤ m ≤ 2n , n ≥ 1} 2n 48 such that r1 ≤ r2 ⇒ G¯r1 ⊆ Gr2 . Observe that the index set is dense in [0, 1]: if s < t in [0, 1], there exists some 2mn such that s < 2mn < t. Define f (x) = inf{r : x ∈ Gr } x ∈ F2 1 x ∈ F2 . Certainly f : X → [0, 1], f (F2 ) = {1}, f (F1 ) = {0}.

### bing-bean topology seminar wisconsin 1965(ISBN 0691080569) by R. H. Bing, Ralph J. Bean

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