Cartesian-closed coreflective subcategories of Tychonoff spaces

Gloria Jean Tashjian

Vol. 81 (1979), No. 2, 547–558

DOI: 10.2140/pjm.1979.81.547

Abstract

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 ≧ℵ₀ 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 ≧ℵ₀ and a finitely productive subfamily 𝒮′ of 𝒮(κ) such that 𝒮⊆𝒮′⊆ 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

Milestones

Received: 19 September 1977

Revised: 6 June 1978

Published: 1 April 1979