Why is the weak topology not more widely defined

Question

I first came across the concept of a weak topology (or initial topology) in the context of functional analysis. It is virtually unmentioned in such standard upper level undergraduate / first year graduate texts as Munkres Topology and Kelley General Topology. Yet it would seem to be a rather ubiquitous and useful concept for constructing important basic topologies.

For example, both the subspace topology and the product topology arise naturally as weak topologies. The latter viewpoint seems especially useful as the standard definition of the product topology to my mind has a somewhat clunky feel.

Given its apparent simplicity and usefulness, why does it not play a more central role in introductions to topology?

Answer 1

In fact, the concept of weak topology occurs in functional analysis: Given a topological vector space $X$, we can retopologize $X$ by giving it the initial topology induced by the family $X'$ of continuous linear functionals living on $X$.

Unfortunately the phrase "weak topology" is not standardized. In the context of CW-complexes it means something completely different: The letter "W" in "CW" stands for weak topology, but here it is a suitable final topology. See the discussion in Confusion about topology on CW complex: weak or final?

Thus "weak topology" and "initial topology" cannot be regarded as synonyms, the interpretation of "weak topology" depends on the context.

Nevertless, the concept of initial topology is quite useful and important. In my opinion it should be treated in good textbooks. However, in older literature (as Kelley and Munkres) it usually does not occur. I guess that thinking in universal properties is a modern approach; it was unusual to do that for a long time. Perhaps it is a matter of taste: The price for the modern aproach is a higher level of abstraction which is not really needed very frequently.