As attached

103318 assignment
Q.no.1
We can see that only is W is complete the pointwise limit lies in W for all elements.(see part ii of the proof where we used Tn tends to T)
Otherwise this proof is not valid.
Hence we can conclude that if W is not complete binary operation from V to W also would not be complete.
1. Let K be the union of all subsets K1, K2…Kn
We
It follows that K is compact.
i.e. union of K1, K2…Kn is also compact set if each Ki is compact
2. Here we are to prove that if n is infinite i.e. when n becomes abnormally large the union may not be compact.
Consider the step
When n becomes very large G0 cannot be an open cover for K.
Hence union need not be compact
Note: Please note that I used T here instead of f as I assumed T is the mapping.
This will not affect the proof
We have, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
Consider...