Re:위상수학 연결성 질문입니다.

[비는아픔]

Before starting, we must accept the following theorem.

Theorem(well-known)
, : connected
If is nonempty, then is connected.

It is easy to check that    

Define and .

Note that if and , both and are nonempty and connected by .
(They are homeomorphic to and respectively.)

Fix .
(They exist because and are proper subsets of and respectively.)

Define and .

Then each for and for is connected by the Theorem.
( and )

Moreover, and are connected by the Theorem, too.
( and )

Now observe that

.

(finally by )

is connected by the Theorem once again.
()

[비는아픔]

안보이시면 말씀하세요 ㅠ.ㅠ

[김수학]

잘보입니다 감사드려요~~