Hi,

I’m studying some category theory by reading Mac Lane linearly and solving exercises.

In question 5.9.4 of the second edition, the reader is asked to construct left adjoints for each of the inclusion functors Top_{n+1} in Top_n, for n=0, 1, 2, 3, where Top_n is the full subcategory of all T_n-spaces in Top, with T_4=Normal, T_3=Regular, etc.

For n=0, 1, 2, it seems to me that I can use the AFT, with the solution set constructed similarly to the one constructed for proving that Haus (=Top_2) is a reflective subcategory of Top (Proposition 5.9.2, p. 135 of Mac Lane).

But I can’t figure out what should I do with the case of n=3, that is, with the inclusion functor Top_4 in Top_3: Top_4 doesn’t even have products, so it seems that I cannot use the AFT.

Is there some direct construction of this left adjoint (by universal arrows, perhaps)? Answers including a reference would be especially helpful.