Sr. No.                            Chapter                                                               Pages 
                          1                      Topological spaces                                                             1 – 18  
                          2                      Bases and subspaces                                                          19 – 28  
                          3                      Special subsets                                                              29 – 46  
                          4                      Different ways of defining topologies                                        47 – 58   
                          5                      Continuous functions                                                        59 – 74  
                          6                      Compact spaces                                                               79 – 96  
                          7                      Connected spaces                                                           97 – 108  
                          8                      First axiom space                                                        109 – 118  
                          9                      Second axiom space                                                       119 – 126  
                         10                      Lindelof spaces                                                           127– 134  
                         11                      Separable spaces                                                         135 – 146  
                         12                      T0 – spaces                                                              147 – 154  
                         13                      T1 – spaces                                                              155 – 174  
                         14                      T2 – spaces                                                              175 – 190  
                         15                      Regular spaces and T  – spaces                                           191 – 202  
                                                                                  3
                         16                      Normal spaces and T  – spaces                                            203 – 218  
                                                                                 4
                         17                      Completely Normal and T  – spaces                                        219 – 228  
                                                                                         5
                         18                      Completely regular and

  – spaces                                      229 – 238  
                                                                                        
                                                                                          
                         19                      Product spaces and Quotient spaces                                       239 – 255  
                                                   
                                       
	

	
        
        
        
        
        
        
        
                            Unit 1 
                       

		

		
                       

		

		
                   §1 Topological spaces:- Definition and examples. 
                   §2 The set of all topologies on X. 
                   §3 Topological spaces and metric spaces 
           
                                                  
        
        
        
        
        
        
        
        
                     
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                                          Unit 1: 

		

 
                                                    

		


                                                                                                        
                      
              §1 Definition and Examples: 
              Definition 1.1: Let X be any non-empty set. A family 
 of subsets of X is called a topology on 
              X if it satisfies the following conditions: 
                
	
  

              
   
      

                

  
	



   

                                                                    
                        If
 is a topology on, then the ordered pair 

   is called a topological space (or T-
              space)  
               
              Examples 1.2: Throughout X denotes a non-empty set. 
              1) 
  
 is a topology on. This topology is called indiscrete topology on and the T-
                        
              space  

  is called indiscrete topological space.  
                                
              2) 
  , (   power set of is a topology on and is called discrete topology on 
                                 
              and the T-space  

  is called discrete topological space.  
              Remark: If  , then discrete and indiscrete topologies on coincide, otherwise they are 
              different.  
                                                                           
              3)  Let  
 
!  then
  
  
  
! 
   and
  
  
   
 
  
   are  topologies 
                                        "                         #
                                        
              on whereas
  
  
   
  is a not a topology on.   
                             $
                                                                           
              4) Let be an infinite set. Define
   %  &  ' 
  then
 is topology on. 
                  (i)   
   …… (by definition of
) 
                       As X – X =  , a finite set,   
  
                  (ii) Let
   
 . If either   or  , then    
 . Assume that  (  and (  .  
                        Then '  is finite and '  is finite. Hence '      '  %  '  is  
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