publication . Article . 1981

A regular space which is not completely regular

A. Mysior;
Open Access
  • Published: 01 Apr 1981 Journal: Proceedings of the American Mathematical Society, volume 81, pages 652-652 (issn: 0002-9939, Copyright policy)
  • Publisher: American Mathematical Society (AMS)
We give a simple example of a regular space which is not completely regular. Every completely regular space is a regular space. The converse is false, however the construction of known counterexamples are quite complicated (compare [4] and [2, Example 2.4.21]). The aim of this note is to provide an elementary construction of a regular noncompletely regular space. The underlying set of our space X is the closed upper half-plane y > 0 plus an additional point a. All points (x, y) with y > 0 are assumed to be isolated. The basic neighborhoods of (x, 0) contain (x, 0) and all but finitely many points from the union of two segments Ix = {(x,y): 0 n} where n = 1, 2 .....
free text keywords: Applied Mathematics, General Mathematics
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