Simplicial Structures in Topology

26 I Fundamental Concepts
every i = 1,...,n, the metric d(−,X � Ai): X → R≥0 is continuous. Hence, the
function
f : X → R≥0 , f = max{d(−,X � Ai) | i = 1,...,n}
is continuous. By Theorem (I.1.26), f (X) is a compact subspace of R and then, by
Theorem (I.1.25), f (X) is closed in R. On the other hand, we note that for any x ∈ X
there exists a certain Ai such that x ∈ Ai and, since X � Ai is closed, d(x,X � Ai) > 0
(see Corollary (I.1.43)); and so, for every x ∈ X, f (x) > 0. Therefore, ℓ = inf{ f (x) |
x ∈ X} > 0. It follows that for every x ∈ X, f (x) ≥ ℓ and then d(x,Ai) ≥ ℓ for some
Ai ∈ A; we conclude that ˚Dℓ(x) ⊂ Ai. �
Exercises
1. Let X be a given set; prove that
is a basis for the discrete topology on X.
B = {{x} |x ∈ X}
2. Let B and B ′ be bases for the topologies A and A ′ on the set X. Provethat
A ′ ⊃ A ⇐⇒ (∀x ∈ X)(∀B ∈ B)x ∈ B,(∃B ′ ∈ B ′ ) x ∈ B ′ ⊂ B,
that is to say, prove that A ′ is finer than A if and only if, for every x ∈ X and every
B ∈ B containing x, there exists B ′ ∈ B ′ containing x and contained in B.
3. Take the following segments of the real line R
(a,b)={x ∈ R | a < x < b},
[a,b)={x ∈ R | a ≤ x < b},
(a,b]={x ∈ R | a < x ≤ b}.
Now take the following sets of subsets of R:
B1 = {(a,b) | a < b},
B2 = {[a,b) | a < b},
B3 = {(a,b] | a < b},
B4 = B1 ∪{B � K | B ∈ B1}, with K = {1/n | n ∈ N}.
Prove that Bi, i = 1,2,3,4, are bases for topologies on R and compare these
topologies.
4. Let X and Y be topological spaces with topologies A and AY , respectively. Let
be the X and Y projection.
Prove that the set
π1 : X ×Y → X
π2 : X ×Y → Y
S = {π −1
1 (U) |U ∈ A}∪{π−1 2 (V ) |V ∈ AY }
is a sub-basis for the product topology on X ×Y.