THE FOLLOWING WEBSITES ARE FOR GAINING UNDERSTANDING

http://pirate.shu.edu/~wachsmut/ira/topo/compact.html

This particular website highlights not only different definitions of compactness within a set but many examples as well.

http://www-history.mcs.st-and.ac.uk/~john/MT4522/Lectures/L21.html

This particular link provides definitions as well as well structured proofs for each definition or corollary they may have

http://www.ucl.ac.uk/~ucahad0/3103_handout_2.pdf

This link speaks about “compactness in metric spaces”, this is more or less to understand a bit as to how and when these compactness is used in Math

THESE ARE THE PROBLEM SETS (INTRODUCTORY)

Prove that ”closed boxes” of the form B = [a1,b1] × ··· × [an,bn] are compact in Rn https://www.math.ksu.edu/~nagy/real-an/1-04-top-compact.pdf

Show that a finite union of compact sets is compact.

Let S be compact and T be closed. Show that S ∩ T is compact

http://www2.hawaii.edu/~robertop/Courses/Math_431/Handouts/HW_Oct_1_sols.pdf

….So all the links highlighted above cannot be used for they are either not licsensed or they have all rights reserved. Therefore these are the alternate links I’ve located thus far:

http://mathoverflow.net/questions/25977/how-to-understand-the-concept-of-compact-space:

licensed under cc by-sa 3.0

http://planetmath.org/examplesofcompactspaces

This link gives examples of compact spaces

https://golem.ph.utexas.edu/wiki/instiki/show/HomePage

this is licensed “As Is” and it appears to be free since is ran through wiki.