A characterization of minimal Hausdorff spaces
| Autor: | Larry L. Herrington, Paul E. Long |
|---|---|
| Rok vydání: | 1976 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 57:373-374 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/s0002-9939-1976-0405355-6 |
| Popis: | This paper gives a characterization of minimal Hausdorff spaces. 1. Preliminary definitions and theorems. A net C: 6D -> X r-converges1 to x0 E X if for each open V containing x0, there exists a d E 6D such that C(Td) C cl (V) [2]. A net C: 6D -* X r-accumulates to x0 E X if for each open V c X containing x0 and for every d E 6D, ?(Td) n cl (V) = 0. Theorem 5 of [1] shows that a Hausdorff space X is minimal Hausdorff if and only if each net in X with a unique r-accumulation point is convergent. A function f: X -4 Y has a strongly-closed graph if for each (x,y) E G(f ) (G(f ) denotes the graph of J) there exist open sets U C X and V C Y containing x and y, respectively, such that (U x cl (V)) n G(f ) = 0 [2]. According to Theorem 7 of [1], each function f: X -4 Y of a topological space X into a minimal Hausdorff space Y with strongly-closed graph is continuous. (Note that Example 3 of [1] shows that the strongly-closed graph condition in Theorem 7 of [1] cannot be relaxed to a closed graph condition.) 2. Main result. Denote by S the class of spaces containing the class of Hausdorff completely normal and fully normal spaces [3]. THEOREM. A Hausdorff space Y is minimal Hausdorff if and only if for every topological space X belonging to 5, each function f: X -> Y with a strongly-closed graph is continuous. PROOF. In view of Theorem 7 of [1], only the sufficiency requires proof. Assume that Y is not minimal Hausdorff. By Theorem 5 of [1] there exists a net f: 6D -* Y with a unique r-accumulation point q E Y such that f does not converge to q. Let xo 4 6D and define X = 6D U {oo}. Then the power set of 6D together with (Td U oo)}Id E 6D} is a base for a topology a on X making (X, a) a fully normal, completely normal Hausdorff space [2], [3]. Define g: X -* Y by g1D = g and g(oo) = q. Using the fact that q is the unique raccumulation point of the net f, it follows that G(g) is strongly-closed. The Received by the editors October 13, 1975. AMS (MOS) subject classifications (1970). Primary 54D20. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|