Vol. 81, No. 2, 1979

Recent Issues
Vol. 294: 1
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
Cartesian-closed coreflective subcategories of Tychonoff spaces

Gloria Jean Tashjian

Vol. 81 (1979), No. 2, 547–558
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 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
Milestones
Received: 19 September 1977
Revised: 6 June 1978
Published: 1 April 1979
Authors
Gloria Jean Tashjian