Edwin E. Moise

The Schröder-Bernstein Theorem

Let A and B be sets. If A ~ B′ ⊂ B, for some B′, then we write A ≤ B. If A ≤ B, but A ≁ B, then we write A < B.

Necessary and Sufficient Conditions for Integrability

Every continuous function [a,b] →ℝ is uniformly continuous. There are three natural proofs of this theorem, using (a) The Nested Interval Theorem, (b) the Heine-Borel Theorem, and (c) the Bolzano-Weierstrass Theorem respectively.

Sets and Functions

We shall use the standard terms and notations of analysis and set theory. (Thus much of the following has already appeared in the first few pages of Analysis.) ℝ is the set of all real numbers, and ℤ is the set of all integers. If A is a set, then x ∈ A means that x belongs to A. If x does not belong to A, then we write x ∉ A. Let S be a set. Then n n

Mappings Between Topological Spaces

left{ {xleft| {x; in ;s;and;left( ldots right)} right.} right}

The Completeness of IR. Uncountable Sets

n ndenotes the set of all elements of S that satisfy the condition (…). Thus the union of two sets A and B is n n

Compactness in Abstract Spaces

A; cup ;B; = ;left{ {xleft| {x; in ;A;or,x, in ;B} right.} right}.