Some properties of the Stone-\v{C}ech compactification.
Hisahiro Tamano · Journal of the Mathematical Society of Japan · 1960
Some properties of the $Stone-\check{C}ech$ compactifcation.105 \S 1. Preliminary.In the first place, we shall state some lemmas concerning reguiariy open sets, which will be used in the following arguments.For the sake of convenience, we shall use the following notations.Let $X$ be a subspace of a topological space $Y$ , and $A$ a set in $X$ , then the closure of $A$ taken in $X$ (or in $Y$ ) will be denoted by $C1_{X}(A)$ (respectively, $C1_{Y}(A)$ ).Similarly, the interior of $A$ will be denoted by $Int_{X}(A)$ or $Int_{Y}(A)$ according as it is taken in $X$ or in $Y$ .A set $F$ in a topological space $X$ is said to be regularly open if the interior of the closure of $F$ is identical with $F$ , that is, $F=Int_{X}(C1_{X}(F))$ .Evidently, it is an open set.LEMMA 1.1.Let $X$ be a topological space and $A$ a regularly open set in $X$ .Let $B$ be an open set which is not contained in A. Then there is an open set $C$ contained in $B$ such that $ C_{\cap}A=\phi$ .PROOF.If $B\subset C1_{X}(A)$ , then $B\subset Int_{X}(C1_{X}(A))=A$ .Therefore we have $Bc ot\subset C1_{X}(A)$ , and $C=B\cap[C1_{X}(A)]^{C}$ is obviously a desired one.LEMMA 1.2.Let $X$ be a dense subspace of a topological space $Y$ and $A$ a set in X.Then $Int_{X}(C1_{X}(A))=Int_{Y}(C1_{Y}(A))_{\cap}X$ .PROOF.The inclusion $Int_{X}(C1_{X}(A))\supset Int_{Y}(C1_{Y}(A))_{\cap}X$ is obvious.There- fore we have only to prove the reversed inclusion.If $p$ is any point of $Int_{X}(C1_{X}(A))$ , then there is in $Y$ an open set $U(p)$ containing $p$ such that $U(p)_{\cap}X\subset C1_{X}(A)=C1_{Y}(A)_{\cap}X$ .It follows that $U(p)\subset C1_{Y}(A)$ and $p$ is there-