Jason (jcreed) wrote,
Jason
jcreed

Ok, so let nTop be the category in which an object is a set X together with n different topologies on X, and in which a morphism f : (X, T1, ... Tn) → (X', T1', ... Tn') is a function X → X' such that (X, Ti) → (X, Ti') is a continuous map between topological spaces for every i.

Then nTop faithfully embeds in mTop iff n ≤ m. That is, it actually is more expressive to consider multiple topological structures.

Proof: Embedding nTop in mTop if n ≤ m is easy: e.g. make the extra topologies discrete. To see that embedding larger n in smaller m doesn't work, notice that any full and faithful embedding would have to preserve the terminal object 1 (the singleton set with each topology being trivial). Furthermore the set of maps from 1 to (X, T1, ... Tn) is equal in cardinality to X, and this must be preserved. So the embedding preserves the cardinality of the underlying set. But some simple combinatorial reasoning yields the conclusion that there are 2n-1(2n + 1) isomorphism classes of objects in nTop whose cardinality is 2.
Tags: math
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