67 Result(s)

Chapter

The Completeness of IR. Uncountable Sets

We now approach the proof that ℝ is uncountable. Since the set Q of all rational numbers is countable, it is clear that any valid proof of the uncountability of ℝ must use the continuity of ℝ. The classical fo...

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

The Existence of Well-orderings. Zorn’s Lemma

The following is analogous to the Unique Finite Sequences Theorem of Section 9.

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

Notations

We shall use the following standard notations of logic and set theory.

Chapter

Absolute Convergence. Rearrangements of Series

If the series Σ|a_i| is convergent, then the series Σa_i is absolutely convergent.

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

Power Series for Elementary Functions

Suppose that f is a function ℝ →ℝ, such that (1) f is differentiable, with f′ = f, (2) f(0) = 1, and (3) there is a series $$\sum\limi...

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

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.

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

Linearly Ordered Spaces

Let [X,<] be a linearly ordered set, in the sense defined in Section 5. We define the induced neighborhood system N = N(<) in the following way. For each a ∈ X, let

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

Map**s Between Topological Spaces

We shall now generalize the definition of a map**, in such a way that it will apply to functions f: X − Y, where [X,O] and [Y,O′] are any topological spaces. The idea that is needed here is brought out in the f...

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

Well-ordering

The following is familiar.

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

The Riemann Integral of a Bounded Function

Let [a,b] be a closed interval in ℝ, let f be a bounded function [a,b] →ℝ, and let M be a bound for f, so that |f(x)| ≤ M for each x ∈ [a,b]. The notations [a,b], f, and M will be used in this sense throughout...

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

Infinite Series

Let a₁, a₂, … be a sequence of real numbers. For each n, let $${A_n}\; = \;\sum\limits_{i\; = \;1}^n {{a_i}} .$$

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

Invertible Functions. Arc-length and Path-length

Let I be an interval in ℝ, let f be a function I → ℝ, and let J = f(I). If f(x) = f(x′) ⇒ x = x′, then f is invertible. If so, there is function f^-1: J → I such that for each x ∈ I, f^-1 (f(x)) = x, and for each y...

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

Power Series

A power series is a series of the form \(\sum\limits_{i\; = \,0}^\infty {{a_i}{x^i}} ,\) where a_i ∈ ℝ for each i. Evi...

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

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 o...

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

Neighborhood Spaces and Topological Spaces

Let G be a collection of sets, let G* be the union of the elements of G, and let X be a set. If X ⊂ G*, then we say that G covers X.

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

Compactness in IRⁿ

In the theory of functions of one real variable, the following is fundamental.

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

The Use of Choice in Existence Proofs

In this section we shall give full discussions and furnish some proofs, because we shall be dealing not with the substance of topology but with various fine points in what one might call Applied Mathematical L...

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

Map**s Between Metric Spaces

We recall the definition of continuity, for a function f:I → ℝ, where I is an interval in ℝ. Let x₀ ∈ I, and suppose that for every ε > 0 there is a ...

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)

Chapter

Connectivity

Roughly speaking, a space X is connected if it is “all in one piece”. Thus if X is a closed interval in ℝ, then X is connected, but if Y is the union of two disjoint closed intervals, then Y is not connected. ...

Edwin E. Moise in Introductory Problem Courses in Analysis and Topology (1982)