On the topology of pointwise convergence on the ... - CARMA

Next, we inductively define an increasing sequence <strong>of</strong> separablesubspaces (F n : n ∈ N) <strong>of</strong> A and countable sets(D n : n ∈ N) in A such that:(i) W ∩ F 1 ≠ ∅;(ii) ⋃ {C x : x ∈ D n ∩ W } ∪ F n ⊆ F n+1 ∈ F for all n ∈ N,where D n is any countable dense subset <strong>of</strong> F n .Note that since <strong>the</strong> family F is rich this construction is possible.Let F := ⋃ n∈N F n and D := ⋃ n∈N D n. Then D = F ∈ Fand ‖ · ‖-diam[f(U)] ≥ ε for every nonempty open subsetU <strong>of</strong> F ∩ W. Therefore, f| F has no points <strong>of</strong> continuityin F ∩ W with respect to <strong>the</strong> ‖ · ‖-<strong>topology</strong>. This however,contradicts Proposition 3.❦ ✂✁