Principles of Mathematics in Operations Research

10

Basic Topology

In this chapter, basic notions in general topology will be defined and the re-

lated theorems will be stated. This includes the following: metric spaces, open

and closed sets, interior and closure, neighborhood and closeness, compactness

and connectedness.

10.1 Metric Spaces

In Rfc, we have the notion of distance:

Ifp= (xi,x 2 ,...,xk)T,q = (yi,V2,---,yk)T, P,Q € Rfc, then

dz(p,q) = y/{xi - yi)^2 + (x 2 - y 2 )^2 + h (xk - Vk)^2

Definition 10.1.1 Let X 7^ 0 be a set. Suppose there is a function
d : X X X => E) = [0,00) with the following properties:
i) d(p, q) = 0 <=> p = q;
ii) d(p,q) = d(q,p), \/p,q;
Hi) d(p,q) < d(p,r) + d(r,q), \/p,q,r [triangle inequality].

Then, d is called a metric (or distance function) and the pair (X, d) is called

a metric space.

Example 10.1.2 Let X ^ 0 be any set. For p,q e X define

d(va) = i

1

'

ifp¥:q

d(P'q) \0, ifp = q

is called the discrete metric.

Definition 10.1.3 Let S be any fixed nonempty set. A function f : S •->• R is
called bounded if f(S) is a bounded subset o/R.