By Sir James H. Jeans, Physics

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Therefore by (18) the two topologies coincide on 5' since the topology Jc is separated and (5',J6) is compact. A metric don fl' for the topology Jc is given according to (6) by the formula d(F,F') := o(F U {co}, F' U {co}). If the sequence (Fn) converges to F in (5'(M),Jc) then the sequence (Fn U I co }) converges to F U { co } in (;r:(M U { co }), o). Therefore, by Theorem 1, FU {co} is the closed limit of the sequence (Fn U {co}) in M U { co }, which implies that F is the closed limit of the sequence (Fn) in M.

Show that the set 5 of closed subsets of M endowed with the topology ::le of closed convergence is compact. ) Problem 4 It has been shown that for a locally compact space (M, ::i) the space (5, ::le) is separated. Show the converse, that is to say, prove that the separated space (M,::i) is locally compact if the space (5,::lc) is sep- arated. Problem 5 Let (M,

To prove this, observe that the space (M,

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