The lattice of weak shift-continuous topologies on C^0 is isomorphic to the product of two copies of the poset of shift-invariant filters with a top element attached, and it contains antichains of size 2^c and chains of size c and t.
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On the lattice of weak topologies on the bicyclic monoid with adjoined zero
The lattice of weak shift-continuous topologies on C^0 is isomorphic to the product of two copies of the poset of shift-invariant filters with a top element attached, and it contains antichains of size 2^c and chains of size c and t.