Topology of the Real Line

Theorem 1 (Open Sets)

The union of any number of open sets in ℝ is open.

Definition (Closed Set)

A subset A of ℝ is called a closed set if its complement A^C is an open set.

Bolzano-Weiestrass Theorem

Let A be a bounded infinite set of real numbers. Then A has at least one limit point.

Heine-Borel Theorem

Let A= [c,d] be a closed and bounded interval, and let G= {Gi : i ∈ I} be a class of open intervals which covers A, i.e., A ⊂ U Gi (i ∈ I). Then G contains a finite subclass {G1, G2, G3,..., Gn} which also covers A, i.e. A ⊂ U Gi (i ∈ I).

Definition (Interior Point)

Let A⊆ ℝ. A point p ∈ A is an interior point A if p belongs to some open interval Sp which is contained in A, i.e., p ∈ Sp ⊆ A.

Definition (Accumulation/Limit Point)

Let A⊆ ℝ. A point p ∈ ℝ is an accumulation point or limit point of A if every open set G containing p contains a point of A different from p, i.e., if G is open and p ∈ G, then A ∩ (G\{0}) ≠ ∅.

Theorem 2 (Open Sets)

The intersection of finite number of open sets in ℝ is open.

Definition (Convergent Sequence)

The sequence (Xn) converges to L if every open set containing L contains almost all, i.e., all but a finite number of the terms of the sequence.

Remark (Open Sets in ℝ)

The set A is open if and only if each of its points is an interior point.

Remarks (Accumulation Points)

The set of all accumulation points of A is called the derived set of A. In symbol, A'.