Re:위상수학 연결성 질문입니다.

작성자비는아픔|작성시간11.07.17|조회수95 목록 댓글 2

Before starting, we must accept the following theorem.

Theorem(well-known)
$X$ , $Y$ : connected
If $X\cap Y$ is nonempty, then $X\cup Y$ is connected.

It is easy to check that $X\times Y - A\times B = ((X-A)\times Y)\cup (X\times(Y-B))$ $\ldots \odot$

Define $W(x) := \{x\}\times Y$ and $Z(y) := X\times \{y\}$ .

Note that if $x\in X-A$ and $y\in Y-B$ , both $W(x)$ and $Z(y)$ are nonempty and connected by $\odot$ .
(They are homeomorphic to $Y$ and $X$ respectively.)

Fix $(a,b) \in (X-A)\times (Y-B)$ .
(They exist because $A$ and $B$ are proper subsets of $X$ and $Y$ respectively.)

Define $U(x):=W(x)\cup Z(b)$ and $V(y):=W(a)\cup Z(y)$ .

Then each $U(x)$ for $x\in X-A$ and $V(y)$ for $y\in Y-B$ is connected by the Theorem.
( $(x,b)\in W(x)\cap Z(b)$ and $(a,y)\in W(a)\cap Z(y)$ )

Moreover, $\displaystyle \bigcup_{x\in X-A} U(x)$ and $\displaystyle \bigcup_{y\in Y-B} V(y)$ are connected by the Theorem, too.
( $\displaystyle (a,b)\in \bigcap_{x\in X-A} U(x)$ and $\displaystyle (a,b)\in \bigcap_{y\in Y-B} V(y)$ )