General TopologyAimed at graduate math students, this classic work is a systematic exposition of general topology and is intended to be a reference and a text. As a reference, it offers a reasonably complete coverage of the area, resulting in a more extended treatment than normally given in a course. As a text, the exposition in the earlier chapters proceeds at a pedestrian pace. A preliminary chapter covers those topics requisite to the main body of work. 
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Book a bit old. The concept of filtre or ultrafiltre is not addressed. Filtres are very powerful tools and using ultrafiltres, we can define hyperreal numbers.
great book
Contents
PRELIMINARIES  1 
UNION AND INTERSECTION  2 
RELATIONS  6 
FUNCTIONS  10 
ORDERINGS  13 
ALGEBRAIC CONCEPTS  17 
THE REAL NUMBERS  19 
COUNTABLE SETS  25 
QUOTIENT SPACES  147 
COMPACTIFICATION  149 
LEBESGUPS COHERING LEMMA  154 
PARACOMPACTNESS  156 
PROBLEMS  161 
UNIFORM SPACES  174 
UNIFORMITIES AND THE UNIFORM TOPOLOGY  175 
UNIFORM CONTINUITY PRODUCT UNIFORMITIES  180 
CARDINAL NUMBERS  27 
ORDINAL NUMBERS  29 
CARTESIAN PRODUCTS  30 
HAUSDORFF MAXIMAL PRINCIPLE  31 
TOPOLOGICAL SPACES  37 
CLOSED SETS  40 
CLOSURE  42 
BASES AND SUBBASES  46 
CONNECTED SETS  53 
PROBLEMS  55 
MOORESMITH CONVERGENCE  62 
SUBNETS AND CLUSTER POINTS  69 
SEQUENCES AND SUBSEQUENCES  72 
CONVERGENCECLASSES  73 
PROBLEMS  76 
PRODUCT AND QUOTIENT SPACES  84 
PRODUCT SPACES  88 
QUOTIENT SPACES  94 
PROBLEMS  100 
EMBEDDING AND METRIZATION  111 
EMBEDDING IN CUBES  115 
METRIC AND PSEUDOMETRIC SPACES  118 
METRIZATION  124 
PROBLEMS  130 
COMPACT SPACES  135 
PRODUCTS OF COMPACT SPACES  143 
LOCALLY COMPACT SPACES  146 
METRIZATION  184 
COMPLETENESS  190 
COMPLETION  195 
COMPACT SPACES  197 
FOR METRIC SPACES ONLY  200 
PROBLEMS  203 
FUNCTION SPACES  217 
COMPACT OPEN TOPOLOGY AND JOINT CONTINUITY  221 
UNIFORM CONVERGENCE  225 
UNIFORM CONVERGENCE ON COMPACTA  229 
COMPACTNESS AND EQUICONTINUITY  231 
EVEN CONTINUITY  234 
PROBLEMS  238 
ELEMENTARY SET THEORY  250 
THE CLASSIFICATION AXIOM SCHEME  251 
EXISTENCE OF SETS  256 
RELATIONS  259 
FUNCTIONS  260 
WELL ORDERING  262 
ORDINALS  266 
INTEGERS  271 
THE CHOICE AXIOM  272 
CARDINAL NUMBERS  274 
282  
293  