connected, 2d countable, normal

a) Reals with the "usual topology." Is there a way to prove this space is normal other than just saying it is normal because every metric space is normal? b) 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 connected? Why 2nd countable?

closure of positive rational numbers

Can you give a brief reason why the closure of the positive rational numbers in each of the topologies below is the way it is indicated. a) Reals with the "usual topology." [0, inf) b) Reals with the "finite complement topology:" U open in X if U - X is finite or is all of X. all reals c) Reals with the "countable complement topology:" U open in X if X - U is countable or is all of X. positive rationals d) Reals with the "lower limit topology:" basis half-closed intervals [a,b) [0, inf) e) Reals with the "upper limit topology:" basis half-closed intervals (a,b] (0, inf) 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

examples of covers of topological spaces

Here is an example of what I would like you to try to do: In space Reals with the "usual topology.", call it X, the covering A={(-n, inf) | n natural number} of X contains no finite subcollection that covers X. Can you do something similar for the following spaces: a) Reals with the "countable complement topology:" U open in X if X - U is countable or is all of X. b) Reals with the "lower limit topology:" basis half-closed intervals [a,b) c) 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, ... } Thank you very much for all your help man... click for more

Sets

Note: C = containment int = interior ext = exterior cl = closure Could you please prove if S C T, then a)int(S) C int(T) b)ext(T) C ext(S) c)cl(S) C cl(T)

Subset Game involving intervals and subintervals

In the following infinite game, Alice and John take turns moving. First, Alice picks a closed interval I₁ of length <1. Then, Bob picks a closed subinterval I₂ which is a subset of or equal to I₁, of length 1/2. Next, Alice picks a closed subinterval I₃ which is a subset of or equal to I₂. The game continues in this way for infinitely many turns. At the endof the game, Alice and Bob have thus picked a sequence of intervals ...(subset of or equal to)I₃(subset of or equal to)I₂(subset of or equal to)I₁, whose intersection consists of a single point x_&infin A referre examines x_{&infinclick for more}