In my book they go through this proof of the following:

An open set

is connected iff for any two points

in

a polygon from

to

lying entirely in

.

We suppose

satisfies this condition and

is not connected to obtain a contradiction.

Pf:

Know

where

are both open and closed,

and

are nonempty.

If we let

and

a polygon from

to

such that

We can assume

Define:

It can be shown that

and

are both open, contradicting the connectedness of

.

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Now, I do not see how S and T are both open. Can anyone give me a hint please? Thank you.

-Sheld