Some Properties of Infinite Sum and Product

-
Click here

Comments

Post a Comment

Popular posts from this blog

TIFR GS 2010 Complete Solution

-
Image
Here is fully solved paper of  TIFR GS 2010  with justification of each and every step, hope you'll enjoy it. For any query feel free to write at  sushantkala786@gmail.com .                                              PART  -  'A' Q.1   Ans.          (c).  phi(60) = 16 Justification   -:     We know that in a cyclic group G of order n generated by a. The order of an element a^(k) is n/(n,k)=n iff (n,k)=1. So the number of generators of G are phi(n). Q.2     Ans.          (a). (a) False,  Consider the direct product Z_3*Z_3*Z_3. (b) True,   Any Abelian group of order 14 must-have elements a and b with respective order 2 and 7 ( Cauchy Theorem ). Then a*b generates the group. (c) True, Same logic as in option (b). (d...
Read more

Compactness in Metric Spaces

-
Image
  The idea of compactness comes from the fact that these are the nice set next to the finite sets. i.e. It shares many properties of finite sets in a topological space. The first thing common between finite and compact sets is, these are closed and bounded. Every sequence has a limit point. A continuous image of a compact set is compact and many more. Compactness has a very close relation to completeness . Let X be a topological space then the definition of compactness is defined in the following way-: X is said to be compact if every open covering of X has a finite sub-covering. Heine-Borel Theorem -: Heine-Borel make our work easy for Euclidean spaces by generalizing that, ‘ K is a compact subset of R if and only if it is closed and bounded. ’ But above theorem doesn’t work well for all metric spaces , for example -: Take N with discrete metric. What is missing here? Completeness? No! Boundedness? No! Closedness? No! Then what? The thing that is missing is totally bou...
Read more