Toplogy Questions

[danzasmack]

My buddy needs help with these and I have no clue. x is the cross product, v is the union.

Anyone have any suggestions?

1.(a) Let X = {p,q} be a space with 2 points p != q (not equal). Given
that {p} is a closed subset of X, list the possible topologies on X.
(b) Are Any of these homeomorphic to each other?
(c) Are any of them metric spaces? Why?
(d) For each of these topologies on X ,what are the continuous functions
from X to [0,1]? From [0,1] to X?

2.(a) For X = S1 v S2, what is pi1(X)? Explain. Does X have some
(non-identity) covering space and can you describe it?
(b) Same for X = S2 v S2
(c) Same for X = S1 x S1
(d) Are any of the pi1's in (a), (b), (c) isomorphic to each other?

[Meromorphic]

What, exactly, does he need help with? It looks to me like he's avoiding learning the basic definitions of topology. For us to do his homework would only allow him to prolong this. /images/graemlins/wink.gif