October 20, 2016
3. Suppose cfw_X1 , X2 , . . . is a sequence in the
Qproduct space X .
a. Show that the sequence converges to X X iff the sequence cfw_ (X1 ), (X2 ), . . .
converges to (X) for each index .
b. Is the statement i
October 13, 2016
1. Suppose X and Y are spaces and y0 Y . Let c : X Y be the constant
function defined by c(x) = y0 . Show that c is continuous.
pf . For any open subset V in Y ,
X, if V contains y0
c (V ) =
, so c
October 12, 2016
Suppose cfw_T is a collection of topologies T on a fixed set X.
a. Show that T is a topology on X.
b. Is T a topoogy? Prove or disprove by a counterexample.
a. Clearly , X T . We have to show that
October 18, 2016
3. Prove the following.
a. A B = A B.
b. A A .
lemma. If A B, then A B. (It is clear since any closed set(e.g. B)
containing B also contains A)
b. Suppose x A , then x A for some . Since A A ,
A A by the abov
Q. Let X = cfw_(x, y) R2 | x2 + y 2 1 and Y = cfw_(x, y) R2 | x2 + y 2
1, x 0 be subspaces of R2 . Show that X and Y are homeomorphic.
pf . Define functions f : X Y, g : Y X by
x + 1 y2
f (x, y) := (
, y), g(x, y) := (2x 1 y 2 , y).
Then g f = IdX