Generating Topologies from Simple Sets

Topology · Axiom Academy

Building complete topologies from simple starting collections In topology, we often start with just a few special sets that we want to be open. But how do we turn that small collection into a complete topology? Let's say we have a set and we want these sets to be open: Watch what happens as we apply the topology axioms to generate the complete topology... First, we need to add all possible unions . A topology must be closed under arbitrary unions! Notice how the generated topology is growing as we add these union combinations. Next, we need all finite intersections . A topology must be closed under finite intersections! We continue this process, checking all possible intersections of our current collection. Finally, we must always include the empty set ∅ and the entire space X . These are required by the topology axioms. This process of generation is fundamental to topology. Often, it's easier to describe a topology by giving a simple collection of sets and letting them generate the full topology. You've seen how topologies can be generated from simple starting collections. This construction process is used throughout topology to build complex structures from simpler parts.

This is the written version of the interactive lesson above. See the full Topology course.