Definition talk:Weak Closure

The notation is not the clearest.

When we are setting up definitions like this, I believe it is important and instructive to specify everything in as much detail as possible. At the moment it is confusing.

We have that $X$ is a topological vector space, and then the $\struct {X, w}$ is a topological vector space with a topology $w$ imposed.

But $X$ itself is properly defined as $\struct {V, \tau}$ where $V$ is a vector space and $\tau$ is a topology.

And then we have $F$ described as being a topological field.

What exactly is the underlying set and what exactly is the underlying vector space and what exactly is the relation between the topology on $X$ and the topology $w$?

I would sort of expect to see:
 * $F$ would be specified as $\struct {F, \tau}$ where $F$ is tacitly understood as being shorthand for the field $\struct {F, +, \times}$, where $F$ is the underlying set of the whole thing.
 * $V$ is a vector space over the field $F$, such that $X = \tuple {V, \tau}$ is the resulting topological vector space over the underlying vector space $V$

and then
 * $\tuple {X, w}$ is the topological vector space over the underlying vector space $V$ and so $X = \struct {V, w}$

or:
 * $\tuple {\tuple {V, \tau}, w}$

which would need further explaining.

There was a lot of discussion way back concerning whether it was always necessary to say something like:
 * Let $T = \struct {S, \tau}$ be a topological space

rather than just taking the topology for granted:
 * Let $T$ be a topological space

but then we've lost track of what the underlying set is.

I still suggest that it is nearly always better to use the full definition whenever the relationship between these objects is subject to misunderstanding. --prime mover (talk) 11:51, 18 February 2023 (UTC)