Separating Points From Closed Sets
If a space X comes equipped with a topology, it is often useful to know whether or not the topology on X is the initial topology induced by some family of maps on X. This section gives a sufficient (but not necessary) condition.
A family of maps {fi: X → Yi} separates points from closed sets in X if for all closed sets A in X and all x not in A, there exists some i such that
where cl denoting the closure operator.
- Theorem. A family of continuous maps {fi: X → Yi} separates points from closed sets if and only if the cylinder sets, for U open in Yi, form a base for the topology on X.
It follows that whenever {fi} separates points from closed sets, the space X has the initial topology induced by the maps {fi}. The converse fails, since generally the cylinder sets will only form a subbase (and not a base) for the initial topology.
If the space X is a T0 space, then any collection of maps {fi} which separate points from closed sets in X must also separate points. In this case, the evaluation map will be an embedding.
Read more about this topic: Initial Topology, Properties
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