Simplicial Structures in Topology

I.1 Topology 3
We need to show that, with this definition, the set U of open sets satisfies properties
A1, A2, and A3 for open sets. Property A1 is easily verified. Let us prove now
that A2 is true: let {Uα | α ∈ J} be a set of elements of U; we want to show that
U = �
α∈J Uα ∈ U. In fact, for each x ∈ U there is α ∈ J such that x ∈ Uα; hence,
there is a B ∈ B such that x ∈ B ⊂ Uα ⊂ U. We prove A3 by induction. Let us
consider two elements U1,U2 ∈ U. If U1 ∩U2 = /0, then U1 ∩U2 ∈ U. Otherwise,
for each x ∈ U1 ∩U2, wemayfindtwosetsB1,B2∈Bsuch that x ∈ B1 ⊂ U1 and
x ∈ B2 ⊂ U2; sinceBis a basis, there exists an element B3 ∈ B such that x ∈ B3 ⊂
B1 ∩ B2. We conclude that x ∈ B3 ⊂ U1 ∩U2 and so, U1 ∩U2 ∈ U. Suppose now that
A3 holds for any intersection of n −1elementsofU. LetU1,U2,...,Un be elements
of U; we write
� �
n� n−1 �
Ui = Ui ∩Un
i=1
i=1
and note that �n−1 i=1 Ui ∈ U by the induction hypothesis; hence, the intersection of
all the given elements Ui belongs to U. We also note that all elements of a basis
are elements of the topology it generates (that is to say, all elements of a basis are
automatically open).
We give now an important example of a topological space which illustrates well
the concepts given so far. Let Rn be the Euclidean n-dimensional space, namely,
the set of all n-tuples x =(x1,...,xn) of real numbers xi. Let d : Rn × Rn → Rn be the function defined in the following manner: for each x =(x1,...,xn) and y =
(y1,...,yn) in Rn ,
� n
d(x,y)= ∑(xi
− yi)
i=1
2 .
This function, called Euclidean distance (or metric) has the following properties:
(∀x,y ∈ R n ) d(x,y)=d(y,x),
d(x,y)=0 ⇐⇒ x = y,
(∀x,y,z ∈ R n ) d(x,z) ≤ d(x,y)+d(y,z).
For every x ∈ R n and every real number ε > 0, we define the n-dimensional open
disk (or open n-disk) with centre x and radius ε as the set
The set
˚D n ε(x)={y ∈ R n | d(x,y) < ε} .
B = { ˚D n ε(x) | x ∈ R n , ε > 0}
is a basis for the Euclidean topology of Rn . In fact, it is evident that condition
B1 above holds; let us prove condition B2: let ˚D n ε(x) and ˚D n δ (y) be two elements
of B with non-empty intersection; for each z in this intersection, consider the real
numbers γ1 = ε − d(x,z) and γ2 = δ − d(y,z); letμbethe minimum of γ1 and γ2.
Then, as we can see from Fig. I.1,
z ∈ ˚D n μ (z) ⊂ ˚D n ε (x) ∩ ˚D n δ (y).