how does one go about showing that a set is open/closed.
e.g the set below is said tobe closed for finite values of T. but not closed for T=
how can we show that?
First, you need to be perfectly clear about the topological space in which this whole problem statement is to be situated.
If it is a metric space you can do the following:
1. To show that is closed (for ) you show that the limit of any convergent sequence of functions also lies in .
2. To show that is not closed (for ) you show that there exists a sequence of functions in , the limit of which does not lie in . (For example: it is easy to find a sequence of functions with limit .)
I'm not sure what the problem is asking for specifically. In general, to show a set is closed, you want to show its complement is open. Are you asking whether and why the set , specifically, is closed? I assume so. (Give enough detail with your problem!)
I suppose the integral is the usual Riemann integral and is Riemann-integrable on every compact subset of . Then is continuous, and for any continuous function , and any , the set is closed. (Prove this! Just apply the definition of continuity.)