Topology Course Notes
Aisling McCluskey and Brian McMaster
August 1997
Chapter 1
Fundamental Concepts
In the study of metric spaces, we observed that:
* many of the concepts can be described purely in terms of open sets,
* openset descriptions are sometimes simpler than metric descriptions, e.g. continuity,
* many results about these concepts can be proved using only the basic properties of open sets (namely, that both the empty set and the underlying set X are open, that the intersection of any two open sets is again open and that the union of arbitrarily many open sets is open).
This prompts the question: How far would we get if we started with a collection of subsets possessing these abovementioned properties and proceeded to define everything in terms of them?
1.1Describing Topological Spaces
We noted above that many important results in metric spaces can be proved using only the basic properties of open sets that
* the empty set and underlying set X are both open,
* the intersection of any two open sets is open, and
* unions of arbitrarily many open sets are open.
We will call any collection of sets on X satisfying these properties a topology. In the following section, we also seek to give alternative ways of describing this important collection of sets.
1.1.1 Defining Topological Spaces
Definition 1 A topological space is a pair (X,T) consisting of a set X and a family T of subsets of X satisfying the following conditions:
* (T1) Æ Î T and X Î T
* (T2) T is closed under arbitrary union
* (T3) T is closed under finite intersection.
The set X is called a space, the elements of X are called points of the space and the subsets of X belonging to T are called open in the space; the family T of open subsets of X is also called a topology for X.

