1. Let S be a subset of and let x ∈ . Prove that one and only one of the following three conditions holds: (a) x ∈ int S (b) x ∈ int ( \S) (c) x ∈ bd S = bd ( \S) 2. Prove the following. (a) An...

1. Let S be a subset of

and let x ∈
. Prove that one and only one of the following three conditions holds:

(a) x ∈ int S

(b) x ∈ int (
\S)

(c) x ∈ bd S = bd (
\S)

2. Prove the following.

(a) An accumulation point of a set S is either an interior point of S or a boundary point of S.

(b) A boundary point of a set S is either an accumulation point of S or an isolated point of S.