I've noticed one classical way of defining certain topologies is to define them as the "weakest" (or coarsest) topology such that a certain set of functions is continuous. For example,
The product topology on $X=\prod X_i$ is the weakest topology such that the canonical projections $p_i : X\to X_i$ are continuous.
The weak*-topology on a Banach space $X$ is the weakest topology such that the evaluation map (the natural isomorphism from $X$ to $X^{**}$, $J(x)(\phi) = \phi(x)$) is continuous.
I have a very poor intuition behind what these "mean". When I look at the definition of a topology, the ones that make the most sense to me are the ones in which the open sets (or at least a base of open sets) are explicitly constructed, as in the Euclidean topology (or a general metric space topology, or a norm induced topology).
I get a little stuck when the definition of the topology is given in some "abstract" sense, where the open sets are "chosen" to satisfy a certain other property. How am I supposed to visualize the open sets in these spaces, or work with them?
If $\tau$ is the weakest topology on $X$ such that $f : X\to Y$ is continuous, is it correct to imagine a base for the open sets to be the preimage of all open sets in $Y$ under $f$? This follows directly from the definition of a continuous function on topological spaces. Is this always the coarsest topology?
Furthermore, what benefit do these topologies provide? What interesting, and potentially theoretically useful, properties do they possess? Why should I care about them?
Best Answer
Yes. For $f$ to be continuous, you need the topology on $X$ to contain all preimages of open sets through $f$. The topology induced by a family $\mathcal T$ of functions is generated by $$ \{f^{-1}(E):\ f\in\mathcal T,\ E\subset Y\ \text{ open }\}. $$ Some reasons why one cares about these topologies are
They often appear naturally, as when considering duals and preduals of normed spaces;
In several cases the topology is coarse enough that some interesting sets become compact (for instance the unit ball in a Banach space, see the Banach-Alaoglu Theorem).