File:Topological space examples.svg

Examples and non-examples of topological spaces, based roughly on Figures 12.1 and 12.2 from Munkres' Introduction to Topology. The 6 examples are subsets of the power set of {1,2,3}, with the small circle in the upper left of each denoting the empty set, and in reading order they are:

• {{}, {1,2,3}} (the trivial topology)
• {{}, {1}, {1,2,3}}
• {{}, {1}, {2}, {1,2}, {1,2,3}}
• {{}, {2}, {1,2}, {2,3}, {1,2,3}}
• {{}, {2}, {3}, {1,2,3}} (not a topology, because {2} and {3} are in the topology but their union {2,3} is not)
• {{}, {1,2}, {2,3}, {1,2,3}} (not a topology, because {1,2} and {2,3} are in the topology but their intersection {2} is not)

25 May 2008(2008-05-25)

Own work

Dcoetzee

See below.