# Closed subset (with respect to weak topology)?

• HMY
In summary, LG is a Hilbert manifold consisting of base point preserving loops, where G is a connected, simply connected Lie group. It is embedded into the Hilbert space L^2[0, 2pi] through a mapping that is weakly continuous. The weak topology on LG is defined as a subset of the weak topology on L^2[0, 2pi], and can be characterized using nets.
HMY
Let LG be the base point preserving loops (it's a Hilbert manifold).
So LG = { f : S^1 -> G s.t. f(0)=1 } where G is a connected, simply connected Lie group.

LG is embedded into the (vector space) Hilbert space L^2[0, 2pi]
given by f |--> g(t) = f '(t)f(t)^-1

Is LG a closed subset of L^2[0,2pi] with respect to the weak topology?
(or is the embedding weakly continuous)?

I figured out the weak topology (reminder here):

Let X = L^2[0,2pi]
Let U(F,b) := { x in X | |F(x)| < b } where b is in R & F is in X^*
So {U(F,b)} are a basis of a neighbourhood of 0 in X.
Thus {x + U(F,b) } are a basis of a neighbourhood of x in X.
ie. the neighbourhoods of an arbitrary x in X are precisely the translates x + W of
neighbourhoods W of 0

So the weak topology on LG is just the subset topology coming from the weak topology
defined on X.

Nets give a nice characterization of the weak topology...

dvs said:
Nets give a nice characterization of the weak topology...

What would this be?

I'm not fluent with nets (yet) but if this characterization is nice enough then it is perhaps easier for me to work with it here, than what I was trying before.

## What is a closed subset with respect to weak topology?

A closed subset with respect to weak topology is a subset of a topological space that contains all its limit points. In other words, every convergent sequence in the subset has its limit point also in the subset.

## How is a closed subset different from an open subset?

A closed subset includes its boundary points, while an open subset does not. In other words, a closed subset is "closed" while an open subset is "open".

## Can a set be both open and closed with respect to weak topology?

Yes, in some cases a set can be both open and closed with respect to weak topology. This is known as a clopen set. An example of this is the empty set, which is both open and closed in any topological space.

## Can a closed subset with respect to weak topology have an infinite number of limit points?

Yes, a closed subset can have an infinite number of limit points, as long as every convergent sequence in the subset has its limit point also in the subset. This is true for any topological space.

## How is the concept of closed subset with respect to weak topology used in mathematics?

The concept of closed subset with respect to weak topology is used in various areas of mathematics, such as analysis, topology, and functional analysis. It is particularly useful in studying the convergence of sequences and the continuity of functions in topological spaces.

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