1. Topological spaces Definition 1.1. We say a family of sets T is a topology if 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 Collections: Mathematics