Spaces whose Pseudocompact Subspaces are Closed Subsets


  • Alan Dow University of North Carolina
  • Jack R. Porter University of Kansas
  • R.M. Stephenson University of South Carolina
  • R. Grant Woods University of Manitoba



Compact, Pseudocompact, Fréchet, Sequential, Product, Extension


Every first countable pseudocompact Tychonoff space X has the property that every pseudocompact subspace of X is a closed subset of X (denoted herein by “FCC”). We study the property FCC and several closely related ones, and focus on the behavior of extension and other spaces which have one or more of these properties. Characterization, embedding and product theorems are obtained, and some examples are given which provide results such as the following. There exists a separable Moore space which has no regular, FCC extension space. There exists a compact Hausdorff Fréchet space which is not FCC. There exists a compact Hausdorff Fréchet space X such that X, but not X2, is FCC.


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Author Biographies

Alan Dow, University of North Carolina

Department of Mathematics

Jack R. Porter, University of Kansas

Department ofMathematics

R.M. Stephenson, University of South Carolina

Department ofMathematics

R. Grant Woods, University of Manitoba

Department of Mathematics


R.W. Bagley, E.H. Connell and J.D. McKnight, Jr., “On properties characterizing pseudo-compact spaces”, Proc. Amer. Math. Soc. 9 (1958), 500–506.

M.G. Bell, “First countable pseudocompactifications”, Topology Appl. 21 (1985), 159–166.

D. Cameron, “A class of maximal topologies”, Pacific J. Math. 70 (1977), 101–104.

Eric K. van Douwen, “The Pixley-Roy topology on spaces of subsets”, Set-theoretic Topology (Papers, Inst. Medicine and Math., Ohio Univ., Athens, Ohio, 1975–1976), 111–134.

Ryszard Engelking, General Topology, Hermann Verlag Berlin, 1989.

L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, New York, 1960.

I. Glicksberg, “The representation of functionals by integrals”, Duke Math. J. 19 (1952), 253–261.

I. Glicksberg, “Stone-Cech compactifications of products”, Trans. Amer. Math. Soc. 90 (1959), 369–382.

Zhou Hao-Xuan, “A conjecture on compact Fréchet spaces”, Proc. Amer. Math. Soc. 89 (1983), 326–328.

E. Hewitt, “Rings of real-valued continuous functions, I”, Trans. Amer. Math. Soc. 64 (1948), 54–99.

Mohammed Ismail and Peter Nyikos, “On spaces in which countably compact sets are closed, and hereditary properties”, Topology Appl. 11 (1980), 281–292.

Peter J. Nyikos, “Classes of compact sequential spaces”, Set Theory and its Applications, Lecture Notes in Mathematics, No. 1401, edited by J. Steprans and S. Watson, Springer-Verlag (1987), 135–159.

Peter J. Nyikos, “Some Moore pseudocompactifications”, Seminar notes at the University of South Carolina (1991).

Jack R. Porter, “Minimal first countable spaces”, Bull. Austral. Math. Soc. 3 (1970), 55–64.

Jack R. Porter and R. Grant Woods, Extensions and Absolutes of Hausdorff spaces, Springer-Verlag, 1987.

Jack R. Porter, R.M. Stephenson, Jr., and R. Grant Woods , “Maximal feebly compact spaces”, Topology Appl. 52 (1993), 203–219.

A. B. Raha, “Maximal topologies”, J. Austral. Math. Soc. 15 (1973), 279–290.

G. M. Reed, “On chain conditions in Moore spaces”, Gen. Topology Appl. 4 (1974), 255–267.

G. M. Reed, “On subspaces of separable first countable T2-spaces”, Fund. Math. 91 (1976), 189–202.

C. T. Scarborough and A. H. Stone, “Products of nearly compact spaces”, Trans. Amer. Math. Soc. 124 (1966), 131-147.

Petr Simon, “A compact Fréchet space whose square is not Fréchet”, Comment. Math. Univ. Carolinae 21 (1980), 749–753.

Petr Simon and Gino Tironi, “First countable extensions of regular spaces”, Proc. Amer. Math. Soc. to appear.

R. M. Stephenson, Jr., “Minimal first countable topologies”, Trans. Amer. Math. Soc. 128 (1969), 115–127.

R. M. Stephenson, Jr., “Minimal first countable Hausdorff spaces”, Pacific J. Math. 36 (1971), 819–825.

R. M. Stephenson, Jr., “Moore-closed and first countable feebly compact extension spaces”, Topology Appl. 27 (1987), 11–28.

A. H. Stone, “Hereditarily compact spaces”, Amer. J. Math. 82 (1960), 900–914.

Toshiji Terada and Jun Terasawa,“Maximal extensions of first-countable spaces”, Proc. Amer. Math. Soc. 85 (1982), 95–99.

S. Watson,“Pseudocompact metacompact spaces are compact”, Proc. Amer. Math. Soc. 81 (1981), 151–152.


How to Cite

A. Dow, J. R. Porter, R. Stephenson, and R. Grant Woods, “Spaces whose Pseudocompact Subspaces are Closed Subsets”, Appl. Gen. Topol., vol. 5, no. 2, pp. 243–264, Oct. 2004.



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