Base (topology) - Weight and Character

Weight and Character

We shall work with Notions established in (Engelking 1977, pp. 12, 127--128). Fix a topological space. We define the weight as the minimum cardinality of a basis; we define the network weight as the minimum cardinality of a network; the character of a point the minimum cardinality of a neighbourhood basis for in ; and the character of to be .

Here, a network is a family of sets, for which, for all Points and open neighbourhoods, there is a for which .

The point of computing the character and weight is useful to be able to tell what sort of bases and local bases can exist. We have following facts:

• obviously .
• if is discrete, then .
• if is Hausdorff, then is finite iff is finite discrete.
• if a basis of then there is a basis of Size .
• if a neighbourhood basis for then there is a neighbourhood basis of Size .
• if is a continuous surjection, then . (Simply consider the -network for each basis of .)
• if is Hausdorff, then there exists a weaker Hausdorff topology so that . So a fortiori, if is also compact, then such topologies coincide and hence we have, combined with the first fact, .
• if a continuous surjective map from a compact metrisable space to an Hausdorff space, then is compact metrisable.

The last fact comes from the fact that is compact Hausdorff, and hence (since compact metrisable spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrisable exactly in case they are second countable. (An Application of this, for instance, is that every path in an Hausdorff space is compact metrisable.)