 
Summary: 1. Topological spaces
Definition 1.1. We say a family of sets T is a topology if
(i) U T whenever U T ;
(ii) F T whenever F T and F is finite.
Note that if T is a topology then = T since T .
Definition 1.2. Let X be a set. A family T of sets is a topology for X if T is a
topology and X = T .
Remark 1.1. Suppose T is a topology for the set X. Since T T and X = T
we have X T . Moreover, if U T then U T X.
Definition 1.3. A topological space is an ordered pair (X, T ) such that X is a
set and T is a topology for X; in this context the members of T are called open
sets and a subset F of X such that X F is open is called closed.
It follows directly from the DeMorgan laws that the intersection of a nonempty
family of closed sets is closed and that the union of a finite family of closed sets is
closed.
Note that and X are always open and closed.
One often says "X is a topological space" so mean that there is T such that
(X, T ) is a topological space.
Definition 1.4. Whenever a Rn
and r is a positive real number we let
