General TopologyAmong the best available reference introductions to general topology, this volume is appropriate for advanced undergraduate and beginning graduate students. Its treatment encompasses two broad areas of topology: "continuous topology," represented by sections on convergence, compactness, metrization and complete metric spaces, uniform spaces, and function spaces; and "geometric topology," covered by nine sections on connectivity properties, topological characterization theorems, and homotopy theory. Many standard spaces are introduced in the related problems that accompany each section (340 exercises in all). The text's value as a reference work is enhanced by a collection of historical notes, a bibliography, and index. 1970 edition. 27 figures. 
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space topology
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It is clear in it's definitions of topology spaces, more especially with the closure and interior properties. It's a must read for all those aspiring mathematicians out there with the knowledge of set theory who like to advance to more rigorous fields of analysis.
Contents
Chapter  1 
Inadequacy of sequences  70 
Compact spaces  116 
23  144 
52  152 
59  159 
Metrizable Spaces 22 Metric spaces and metrizable spaces  161 
Metrization  163 
The homotopy relation  222 
The fundamental group  227 
FMS  233 
Uniform Spaces 35 Diagonal uniforrnities  238 
Uniform covers  244 
Uniform products and subspaces weak uniforrnities  251 
Uniforinizability and uniform metrizability  255 
Complete unifonn spaces completion  260 
70  170 
Complete metric spaces  179 
The Baire theorem  188 
Connectedness 26 Connected spaces  191 
85  192 
Pathwise and local connectedness  199 
Continua  203 
Totally disconnected spaces  213 
The Cantor set  218 
3l Peano spaces  219 
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Common terms and phrases
algebraic axiom of choice Baire Banach space basic nhood Cauchy closed sets closed subset closure cluster point collection compact metric space compact spaces compact subset compactiﬁcation completely metrizable completely regular continuous functions continuous map converges deﬁned Deﬁnition denote dense elements embedded equivalent example Exercise f is continuous ﬁlter ﬁnd ﬁrst countable ﬁxed function f Hausdorff space Hausdorﬂ hence homeomorphic homotopy iﬂ inﬁnite intersection interval Lemma Let f Lindelof locally compact map f Math metrization theorem nhood nhood base nonempty normal spaces one—one open cover open set paracompact space pointwise product space product topology Proof proved pseudometric pseudometric space quotient result retract second countable Section separable subbase subspace Suppose T,space theory topological space topology induced totally bounded Tychonoff space ultraﬁlter uniform cover uniform space uniformly continuous union Urysohn usual topology weak topology