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 compl ...continues
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 ...continues
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 t ...continues
(4) (a) Let I1,I2,I3... be open intervals and let J be a closed interval and let J be a closed inteval. Let lk be the length of Ik, and let L be the length of J....Please see the attachement
Could someone please provide me with the solution to QA1 of the attatched exam paper to aid my revision.
Topology - Past exam paper QA2 & A3
Could someone please provide me with the solution to QA2 & A3 of the attatched exam paper to aid my revision.
Topology : Homomorphism (Question B5)
Please see the attached file for the fully formatted problems. B5. (a) Define a homomorphism between topological spaces X and Y. Define what is meant by a topological invariant. (b) State what it means for a map f X -•> Y to be open. Show that a continuous open bijection is a homomorphism. (c) (i) Recall that Fr E, the fron ...continues
Topology : Connected Spaces and Explosion Point
Please see the attached file for the fully formatted problems. B6. (a) Define what it means for a topological space to be connected. (b) Suppose that A and B are subspaces of a topological space X, and that U C A fl B is open in both A and B in the relative topologies. Show that U is open in A U B in the relative topology. ...continues
