Vol. 81, No. 2, 1979

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Cartesian-closed coreflective subcategories of Tychonoff spaces

Gloria Jean Tashjian

Vol. 81 (1979), No. 2, 547–558

Let 𝒮 be a class of spaces in the category of Tychonoff spaces and let co(𝒮) be its coreflective hull in that category, with coreflector c.

Let τ be the topology of uniform convergence on the set of continuous maps C(X,Y ).

For κ 0 let 𝒮(κ) be the collection of all Tychonoff spaces which are pseudo-κ-compact and m-discrete for every m < κ.

Theorem. The following are equivalent:

  1. co(𝒮) is cartesian-closed and the exponential objects for S ∈ 𝒮 and Y co(𝒮) are the spaces cτC(S,Y ).
  2. The projection π : c(S × T) S is z-closed for each S,T ∈𝒮.
  3. Either co(𝒮) is the category of discrete spaces, or there exists κ 0 and a finitely productive subfamily 𝒮′ of 𝒮(κ) such that 𝒮𝒮′co(𝒮).

Furthermore, if 𝒮 is map-invariant, then (a) implies that all spaces in 𝒮 are pseudocompact.

Several examples are given.

Mathematical Subject Classification 2000
Primary: 54B30
Secondary: 54B10, 54E15
Received: 19 September 1977
Revised: 6 June 1978
Published: 1 April 1979
Gloria Jean Tashjian