And the founder, having shod a plough with a brazen ploughshare, and having yoked to it a bull and a cow, himself drove a deep furrow round the boundary lines, while those who followed after him had to turn the clods, which the plough threw up, inwards towards the city, and suffer no clod to lie turned outwards.

Plutarch • Life of Romulus

Chapter 1. Topological Spaces

1.1. Topologies and Neighborhoods

A topology is a family of sets which satisfies the two conditions:  the intersection of any two members of is a member of and the union of the members of each subfamily of is a member of   The set is necessarily a member of because is a subfamily of itself, and every member of is a subset of   The set is called the space of the topology and is a topology for   The pair is a topological space.  When no confusion seems possible we may forget to mention the topology and write “ is a topological space.”  We shall be explicit in cases where precision is necessary (for example if we are considering two different topologies for the same set ).

The members of the topology are called open relative to or -open, or if only one topology is under consideration, simply open sets.  The space of the topology is always open, and the void set is always open because it is the union of the members of the void family.  These may be the only open sets, for the family whose only members are and the void set is a topology for   This is not a very interesting topology, but it occurs frequently enough to deserve a name;  it is called the indiscrete (or trivial) topology for and is then an indiscrete topological space.  At the other extreme is the family of all subsets of which is the discrete topology for (then is a discrete topological space).  If is the discrete topology, then every subset of the space is open.  (Kelley, p. 37).