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About LinkBackshow 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?
:\int_0^T f(x) dx = 1\})
First, you need to be perfectly clear about the topological space in which this whole problem statement is to be situated.Originally Posted by mtmath
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?
If it is a metric space you can do the following:
1. To show thatis closed (for
) you show that the limit
of any convergent sequence of functions
also lies in
2. To show that) you show that there exists a sequence of functions
in
.)
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. Then
is continuous, and for any continuous function
, and any
, the set
is closed. (Prove this! Just apply the definition of continuity.)
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