Question

If X, Y are topological spaces and f : X → Y we call the graph of f the set Γf = {(x, f(x)); x ∈ X} which is a subset of X × Y.

If X and Y are metric spaces and f is a continuous function prove that the graph of f is a closed set.

Answer #1

Suppose that f : X → Y and g: X → Y are
continuous maps between topological spaces and that Y is Hausdorff.
Show that the set A = {x ∈ X : f(x) = g(x)} is closed in X.

Let (X, dX) and (Y, dY ) be metric spaces and let f : X → Y be a
continuous bijection. Prove that if (X, dX) is compact, then f is a
homeomorphism

Let X and Y be metric spaces. Let f be a continuous function
from X onto Y, that is the image of f is equal to Y. Show that if X
is compact, then Y is compact

Please prove the following theorem:
Suppose (X,p) and (Y,b) are metric spaces, X is compact, and
f:X→Y is continuous.
Then f is uniformly continuous.

If X and Y are discrete topological spaces, how do I prove X x Y
is also discrete?
How would I prove the converse is true, could I ? Very interested.
Please explain.
Like to please break down into steps that would make sense. Not
sure if I have to use facts of product topology and discrete
topology and how to use them.

Let X, Y be metric spaces, with Y complete. Let S ⊂ X and let f
: S → Y be uniformly continuous. (a) Suppose p ∈ S closure and (pn)
is a sequence in S with pn → p. Show that (f(pn)) converges in y to
some point yp.

Let (X1,d1) and (X2,d2) be metric spaces, and let y∈X2. Define
f:X1→X2 by f(x) =y for all x∈X1. Show that f is continuous.
(TOPOLOGY)

Prove that if f: X → Y is a continuous function and C ⊂ Y is
closed that the preimage of C, f^-1(C), is closed in X.

Let f be a continuous from a topological space X to the reals.
Let a be in the reals and let A = {x in X : f(x)=a} Show that A is
closed inX.

1.- Prove the intermediate value theorem: let (X, τ) be a
connected topological space, f: X - → Y a continuous transformation
and x1, x2 ∈ X with a1 = f (x1), a2 = f (x2) ( a1 different a2).
Then for all c∈ (a1, a2) there is x∈ such that f (x) = c.
2.- Let f: X - → Y be a continuous and suprajective
transformation. Show that if X is connected, then Y too.

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