# Topology, 2/E by James Munkres

By James Munkres

For a senior undergraduate or first yr graduate-level path in creation to Topology. applicable for a one-semester direction on either common and algebraic topology or separate classes treating every one subject separately.
This textual content is designed to supply teachers with a handy unmarried textual content source for bridging among normal and algebraic topology classes. separate, distinctive sections (one on common, element set topology, the opposite on algebraic topology) are each one compatible for a one-semester path and are dependent round the similar set of simple, center subject matters. not obligatory, self reliant themes and functions may be studied and constructed intensive reckoning on direction wishes and preferences.

Table of Contents

I. normal TOPOLOGY.
1. Set thought and Logic.

2. Topological areas and non-stop Functions.

three. Connectedness and Compactness.

four. Countability and Separation Axioms.

five. The Tychonoff Theorem.

6. Metrization Theorems and Paracompactness.

7. entire Metric areas and serve as Spaces.

eight. Baire areas and measurement Theory.

II. ALGEBRAIC TOPOLOGY.
nine. the basic Group.

10. Separation Theorems within the Plane.

11. The Seifert-van Kampen Theorem.

12. type of Surfaces.

13. category of masking Spaces.

14. functions to team Theory.

Index.

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Example text

C) {x x is an integer for all I 100}. I (d) {xIx2=x3}. §6 Finite Sets Finite sets and infinite sets, countable sets and uncountable sets, these are types of sets that you may have encountered before. Nevertheless, we shall discuss them in this section and the next, not only to make sure you understand them thoroughly, but also to elucidate some particular points of logic that will arise later on. First we consider finite sets. Recall that if n is a positive integer, we use to denote the set of positive integers less than n; it is called a section of the positive integers.

Proof The function f : —÷ of Z÷ with a proper subset of itself. 5. — (1} defined by f(n) = n + 1 is a bijection U The cardinality of a finite set A is uniquely determined by A. Proof Let m

We leave it to the exercises to show that A has the least upper bound property if and only if it has the greatest lower bound property. Consider the set A = (—1, 1) of real numbers in the usual order. Assuming the fact that the real numbers have the least upper bound property, it follows that EXAMPLE 13. 28 Ch. 1 Set Theory and Logic the set A has the least upper bound property. For, given any subset of A having an upper bound in A, it follows that its least upper bound (in the real numbers) must be in A.

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