Vtyjkyuk

Let $Int(Y)$ denote the interior of $Y,$ and let $overlineX$ denote the closure of $X.$ How do you show that

$$Int(Y) cup Int(Z) subseteq Int(Y cup Z)?$$ Can you give an example?

I know this is the complement of $overline Ycap Z subset overlineY cap overlineZ$ but it's difficult for me to understand the relationship between these two.

If $tau$ is the topology of $X$,
$$xin Int Ycup Int Zimpliesexists U(Uintauwedge xin Uwedge (Usubset Yvee Usubset Z))$$
$$implies exists U(Uintauwedge xin Uwedge Usubset Ycup Z)implies xin Int(Ycup Z)$$

Let $V(Y)$ denote the family of all open subsets of the set $Y.$ (This is not a standard notation.)

(1).Def'n: Int$(Y)=cup V(Y).$

(2). Since a union of a family of open sets is open, Int$(Y)$ is open. Since every member of $V(Y)$ is a subset of Y we have Int$(Y)=cup V(Y)subset Y.$ So Int$(Y)$ is an open subset of Y. Therefore by (1) we have Int$(Y)in V(Y).$

(3). If $W$ is an open subset of $Y$ then $Win V(Y) implies Wsubset cup V(Y)=$Int$(Y)in V(Y).$ So Int$(Y)$ is the largest open subset of $Y.$ That is, any member of $V(Y)$ is a subset of Int$(Y).$

(4). Int$(X)$ and Int$(Y)$ are open sets with Int$(X)subset X$ and Int$(Y)subset Y. $ So Int$(X) cup$Int$(Y)$ is an open subset of $X cup Y .$ That is, Int$(X)cup$Int$(Y)in V(Xcup Y). $ Therefore by (3) we have $$ text Int(X)cup text Int(Y)subset text Int(Xcup Y).$$

Examples.[1]. In the space $mathbb R$ let $X=mathbb Q$ and $Y=mathbb R$ $X.$ Then Int$(X)=$Int$(Y)=emptyset$ ................but Int$(Xcup Y)=$Int$(mathbb R)=mathbb R.$

[2].In $mathbb R$ let $X=(0,1]$ and $Y=[1,2).$ Then Int$(X)cup$Int$(Y)=(0,1)cup (1,2)$ ........................and Int$(Xcup Y)=(0,2).$