Simplicial Structures in Topology

I.2 Categories 27
5. Prove that a function f : Z → X ×Y is continuous if and only if its components
f1 = π1 f and f2 = π2 f are continuous.
6. Amap f : X → Y is open if and only if, for every U ∈ A), f (U) is open in Y .
Show that the projection maps
are open.
π1 : X ×Y → X
π2 : X ×Y → Y
7. Let X and Y be topological spaces and let q: X → Y be a surjection; give Y the
quotient topology relative to q. Prove that, for any topological space Z, any function
g: Y → Z is continuous if and only if gq: X → Z is continuous.
8. Endow R with the Euclidean topology and let f : R → R be a given function.
Prove that the following statements are equivalent.
(∀U ⊂ R |U open )( f −1 (U) open);
(∀x ∈ R)(∀ε > 0)(∃δ > 0)|x − y| < δ ⇒|f (x) − f (y)| < ε.
9. Prove that if A is closed in Y and Y is closed in X,thenA is closed in X.
10. Prove that if U is open in X and A is closed in X, thenU � A is open in X and
A �U is closed in X.
11. Show that a discrete space is compact if and only if it is finite.
12. Let A and B be two non-empty subsets of Rn .Thedistance between A and B is
defined by
d(A,B)=inf {d(a,b) | a ∈ A,b ∈ B}.
Prove that if A � B = /0, A and B are closed, and A is bounded, then there exists a ∈ A
such that
d(A,B)=d(a,B) > 0.
I.2 Categories
I.2.1 General Ideas on Categories
A category C is a class of objects together with two functions, Hom and Composition,
satisfying the conditions:
Hom: it assigns to each pair of objects (A,B) of C asetC(A,B); anelement
f ∈ C(A,B) is a morphism with domain A and codomain B;