On nearly discrete spaces
1950; Institute of Mathematics of the Czech Academy of Sciences; Volume: 075; Issue: 2 Linguagem: Inglês
10.21136/cpmf.1950.120774
ISSN1802-114X
Autores Tópico(s)Advanced Topology and Set Theory
ResumoNearly discrete and nearly (R) discrete spaces investigated in this note are, roughly speaking, Hausdorff (respectively, regular) spaces the topology of which cannot be modified without either some open set becoming non-open or some non-isolated point becoming isolated.It is immediately seen that discrete (i.e. Buch that every subset is open) spaces as well as spaces investigated by E. HEWITT [1] *) and by the present author [2] and called maximal (maximal completely regular) in [1], minimal (JK-minimal) in [2] are nearly [respectively, nearly (R)] discrete.In this note some properties of nearly discrete and nearly (R) discrete spaces are examined and a close relation is established between nearly (R) discrete spaces and the Cech (bi)compactification of discrete spaces (for instance, every countable nearly (R) discrete space may be imbedded into the Cech compactification of natural numbers).We begin with some preliminary definitions and lemmas.All spaces considered are Hausdorff topological spaces.The letters P, S always denote spaces.Sapping means a continuous transformation; function means a real*valued function.; Definitions.A space P is called dense-in-itself if it contains no isolated point, dispersed Hit contains no non-void dense-in-itself subspace.A set Q C P is called relatively dense-in-itself (in P), abbreviated r. d., if it contains no isolated (in Q) points except those isolated already in P, relatively almost dense-in-iiself t abbreviated almost r.d., if the set of all re € ©which ire isolated in Q without being isolated in P is countable (== finite or" countable infinite).A one-to-one mapping \p of a space P x into P 2 is called an i-mapping if tp(x) is an isolated point in P 2 whenever x is isolated in P v If 9? is an t-mapping of P x onto P a , theii P, is said to be m i-image of P v ;-^srn i*k;.. f /....-• Lemma 1.In any space P, the sum of an arbitrary coltectidn of !--r.d. sets is r.d.y every open set is r.d.t the intersection of an open set and a r.d.*) Numbers in brackets refer to tha list at the end of the paper.V
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