Usage of "weakly" when dealing with properties in the weak topology

Question

Sometimes, when I read about topologies, for example the weak topology, they usually say "converges weakly", or "weakly continuous", "weakly open", etc., something similar, without mentioning what they mean by using "weakly" in this context.

Is the term "weakly" added beside a property with it is associated with the weak topology, or how exactly? Do people say, "--- is something weakly" if "--- is something with respect to the weak topology"? (Note, they first talk about the weakly topology, thereafter they say something "weakly", not the converse direction.)

The same goes for other kind of topologies.

Answer 1

"Weakly" is not always associated with a weak topology. For example, there are weakly prime numbers with no weak topology in sight.

Answer 2

If the property relates to a topic in analysis (e.g. convergence, continuity, compactness, etc.) then most of the time "Weakly X" refers to "X under the weak topology". Thematically, this means "watch out if your space is infinite-dimensional" because the weak and strong topologies do not necessarily coincide in that context.

In response to a comment above: while it is very common to use "weakly" in this format, I've never seen someone use a different adjective, e.g. I have never seen someone use "discretely converges" to say that something "converges in the discrete topology".