Question

In
Metric Spaces, prove d(f,g)=d(g,f)

Answer #1

First, note the definition of a metric on a set.

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 E and F be two disjoint closed subsets in metric space
(X,d). Prove that there exist two disjoint open subsets U and V in
(X,d) such that U⊃E and V⊃F

Suppose d and d 0 are both metrics on X and that the metric
spaces (X, d) and (X, d0 ) have the same open sets. Show, the
sequence (an) from X converges in (X, d) if and only if it
converges in (X, d0 ) and then to the same limit.

The product of two metric spaces (Y, dY ) and (Z, dZ) is the
metric space (Y × Z, dY ×Z), where dY ×Z is defined by dY ×Z((y,
z),(y 0 , z0 )) = dY (y, y0 ) + dZ(z, z0 ).
Assume that (Y, dY ) and (Z, dZ) are compact. Prove that (Y × Z,
dY × dZ) is compact.

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

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)

5.1.5. Suppose V1, V2, W are vector spaces over F. Prove that f
: V1 × V2 → W is the zero map if and only if f is both linear and
bilinear.

is about metric spaces:
Let X be a metric discret space show that a sequence x_n in X
converge to l in X iff x_n is constant exept for a finite number of
points.

Prove the following
Let f : A → B then, for all D, E ⊆ A and for all G, H ⊆ B we
have
f-1(G ∪ H) = f-1(G) ∪
f-1(H)

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