Topological spaces

Which of the following topological spaces is normal? Please give a proof why or why not. Thank you very much. a) Reals with the "usual topology." b) Reals with the "finite complement topology:" U open in X if U - X is finite or is all of X. c) Reals with the "countable complement topology:" U open in X if X - U is countable or is all of X. d) Reals with the "lower limit topology:" basis half-closed intervals [a,b) e) Reals with the "upper limit topology:" basis half-closed intervals (a,b] f) Reals with the "K-topology:" basis consists of open intervals (a,b) and sets of form (a,b) - K where K = {1, 1/2, 1/3, ... }

Connected topological spaces.

Please show why (briefly) each of the following top. spaces is or is not connected as indicated. Thank you. a) Reals with the "usual topology." Why connected? b) Reals with the "finite complement topology:" U open in X if U - X is finite or is all of X. Why connected? c) Reals with the "countable complement topology:" U open in X if X - U is countable or is all of X. Why connected? d) Reals with the "lower limit topology:" basis half-closed intervals [a,b). Why not connected? e) Reals with the "upper limit topology:" basis half-closed intervals (a,b]. Why not connected? f) Reals with the "K-topology:" basis consists of open intervals (a,b)and sets of form (a,b) - K where K = {... click for more

Determine 2nd countability of given topological spaces.

Which of the following topological spaces is 2nd countable? Please show how you obtained your result. Thank you. a) Reals with the "usual topology." b) Reals with the "finite complement topology:" U open in X if U - X is finite or is all of X. c) Reals with the "countable complement topology:" U open in X if X - U is countable or is all of X. d) Reals with the "lower limit topology:" basis half-closed intervals [a,b) e) Reals with the "upper limit topology:" basis half-closed intervals (a,b] f) Reals with the "K-topology:" basis consists of open intervals (a,b) and sets of form (a,b) - K where K = {1, 1/2, 1/3, ... }

Compact, Regular topological spaces.

a) Reals with the "finite complement topology:" U open in X if U - X is finite or is all of X. Why compact? Why not regular? b) Reals with the "countable complement topology:" U open in X if X - U is countable or is all of X. Why not compact? Why not regular. c) Reals with the "K-topology:" basis consists of open intervals (a,b) and sets of form (a,b) - K where K = {1, 1/2, 1/3, ... } Why not compact? Why not regular? Thank You

normal implies completely regular

A Hausdorff space is said to be completely regular if for each pt. x in X and closed set C with x not in C, there exists a continuous function f: X --> {0,1} s.t. f(x)=0 and f(C)={1}. Show that if a space is normal, it is completely regular. How do I use Urysohn's lemma along with Hausdorffiness to show this. Thank You