# Topology of the Reals proofs

• Feb 11th 2009, 10:55 PM
noles2188
Topology of the Reals proofs
I need help proving the following statements. I've been doing this homework for 6 hours and I am finally burnt out. Any help is greatly appreciated.

Let R denote the set of all reals.

1) Let S be a subset of R and let x be an element of R. Prove that one and only one of the following three conditions holds:
a) x is an element of the set of all interior points S (int S)
b) x is an element of int (R\S)
c) x is an element of the set of all boundary points S (bd S) = bd (R\S)

2) Let S be a bounded infinite set and let x = sup S. Prove: If x is NOT an element of S, then x is an element of S' (set of all accumulation points of S).

3) Prove: bd S = (cl S) intersection [cl (R\S)].

Thank you. Any help would be awesome.
• Feb 12th 2009, 03:12 AM
Plato
Quote:

Originally Posted by noles2188
I need help proving the following statements. I've been doing this homework for 6 hours and I am finally burnt out. Any help is greatly appreciated.

Let R denote the set of all reals.

1) Let S be a subset of R and let x be an element of R. Prove that one and only one of the following three conditions holds:
a) x is an element of the set of all interior points S (int S)
b) x is an element of int (R\S)
c) x is an element of the set of all boundary points S (bd S) = bd (R\S)

3) Prove: bd S = (cl S) intersection [cl (R\S)].

Problems 1 & 3 are really about using the definition of boundary points.
“p is a boundary point of S if and only if each open set containing p contains a point of and a point not in S”.
From that statement it should be clear that $\displaystyle p \in \beta (S)\; \Rightarrow \;p \in \beta (S^c )$ every point is the boundary of S is in the boundary of S complement.
• Feb 12th 2009, 02:30 PM
aliceinwonderland
Quote:

Originally Posted by noles2188
I need help proving the following statements. I've been doing this homework for 6 hours and I am finally burnt out. Any help is greatly appreciated.

Let R denote the set of all reals.

1) Let S be a subset of R and let x be an element of R. Prove that one and only one of the following three conditions holds:
a) x is an element of the set of all interior points S (int S)
b) x is an element of int (R\S)
c) x is an element of the set of all boundary points S (bd S) = bd (R\S)

2) Let S be a bounded infinite set and let x = sup S. Prove: If x is NOT an element of S, then x is an element of S' (set of all accumulation points of S).

3) Prove: bd S = (cl S) intersection [cl (R\S)].

Thank you. Any help would be awesome.

Another way to show (1) and (3) is that
"int S" is the largest open set contained in S and "int R\S" is the largest open set contained in R\S. Since a union of open sets is an open set, "int S" $\displaystyle \cup$ "int R\S" is an open set.
Now, there is a closed set which is R\("int S" $\displaystyle \cup$ "int R\S"). You need to show that the closed set is bd(S) using R\("int S" $\displaystyle \cup$ "int R\S").

Thus, "int S" , "int R\S", and "bd(S)" are disjoint sets whose union is R.

For (2), the Bolzano-Weierstrass theorem states that "every bounded infinite subset of R has a limit point". If S is a closed set, limit points are in S. If S is an open set, limit points are either in S or S'.