Generalized Realcompactness

Generalized
Realcompactness

This site contains a table of all the known generalizations of realcompactness, as well as a summary of the relationships between them. This is intended to be a dynamic site, with new generalizations of realcompactness being added as they are discovered. It is hoped that members of the community will assist in the maintainance of this site by making submissions. A submission can range from a completely new result or generalization, to a better citation for something that already exits in the table. In any case, a citation should accompany all submissions. Comments, of course, are also welcome.

Note: This site is in desperate need of revision. The above diagram is relatively new, but it contains generalizations of realcompactness that have not been integrated into the rest of this site. Among the "new" generalizations that have been added to the diagram include those offered by Isiwata of early 80's vintage: wa-realcompact, (α), cb*, weak cb*, OPO, OPC, WOPC, ZC, and CPC. Fletcher and Kunzi are responsible for PF-compact (1983), and Milovancevic for R-space (1984). This whole site will be remodeled during the 2001-2002 academic year.

A brief history of weak forms of realcompactness is published in the Topology Atlas. The definitions of the properties in the table below have also been collected into a website.

The table below is an expanded version of one that appeared in a 1980 article of Blum and Swaminathan. The symbol "+" appears in the cell at the intersection of row A and column B if A implies B holds. Each capital letter that appears in a cell is an example of a space satisfying property A but not B. M is the Mrowka space [Mro], My isthe Mysior space [Mys], $Ψ$ the Isbell space [GJ,5I], S theTychonoff Corkscrew [JM,7.3], P the non-realcompact P-space [GJ, 9L], W the countable ordinals [GJ,5.12], H the Mack-Johnson space[MJ,p.240], D the Dieudonne Plank [Ste,p.108], R Rudin's Dowker space[Rud], U any discrete space of Ulam-measurable cardinality, Q the space $Q × ω$ ₁ , B the Blair-van Douwen space[BvD,1.15], and Sz the Swardson-Szeptycki space [SS,2.8(2)]

real comp unif hyper- strongly iso
real + +[GJ,15.20] + + +
comp unif U + +[Dy1,3.1] + +
hyper- U,P P[BLS,3.9] + + +
strongly U,P,D P,D D[BLS,3.9] + +
iso M,My,Ψ,
U,P,D Ψ,D Ψ,D Ψ[BLS,3.9] +
almost M[W,16.12],
My[My],
D[PW,6U] D D +
weakly Bc M,My,
Ψ,D D,Ψ D,Ψ Ψ +
a-realcompact
closed complete [GP,8.12] M,My,Ψ,
D,R[Sim] D,Ψ D,Ψ Ψ +
pure M,My,Ψ,
D,R D,Ψ D,Ψ Ψ +
$δ$ -neat M,My,Ψ,
D,R D D +
neat M,My,Ψ,
D,R,P D,P,Ψ D,Ψ Ψ +[Sak,2.6]
c-real M,My,H,
S,B,D B,D B,D B B[SS,2.8]
p-real M,My,H,U
S,B,D,Sz B,D,Sz B,D,Sz B,Sz Sz[SS,2.8],B
$ψ$ -com M,My,H,
P,D P,D D
$μ$ -com M,My,H,
P,W,D P,D D
$η$ -com
nearly real [JM,6.1] M,My,H,
P,S,Q,U
B,D,Sz Q,B,D,
Sz Q,B,D,
Sz Q,B,Sz Sz,B
$λ$ -com H,P,W P

almost weakly Bc a-real pure $δ$ -neat neat
real +[F,Thm 10] +[GP,14.11] + + + +
comp unif
hyper- P P
strongly P P
iso Ψ,R,P R,P P P
almost + +[RR,2.1] +[Dy2,1.6] + + +
weakly Bc Ψ[RR,Ex3] + +[GP,6.13] + + +
a-realcompact
closed complete [GP,8.12] Ψ[BS,3.2],
R R[Sim] + + + +
pure R,Ψ R + + +
$δ$ -neat R,Ψ R + +
neat P,R,Ψ P,R P P[Sak,3.8] +
c-real H,S,B H,S,B H[BS,4.0],
S[BS3.3],
B B B B
p-real H,S,B,
Sz H,S,B,
Sz H,S,B,
Sz B,Sz B,Sz B,Sz
$ψ$ -com H,P H,P H[BS,4.0],
P[BS,3.4]
$μ$ -com H,P,W H,P,W H,P,
W[BS,3.5]
$η$ -com
nearly real [JM,6.1] H,P,S,
Sz H,P,S,
Sz H,P,S,
Sz Sz Sz Sz
$λ$ -com H,P,W H,P,W H[BS,4.0],
P[BS,3.4],
W[BS,3.5]

c-real p-real psi-com mu-com eta-com lambda-com
real + + + + + +
comp unif + +
hyper- P + +
strongly P +[SS,2.6] +
iso P,Ψ Ψ Ψ Ψ Ψ M,Ψ
almost +[Dy2,3.3] + +[BS,2.1] + + M[BS,3.1]
weakly Bc Ψ Ψ Ψ Ψ Ψ M,Ψ
a-realcompact
closed complete [GP,8.12] Ψ Ψ Ψ Ψ[BS,3.2] Ψ[BS,3.2] M,
Ψ[BS,3.2]
pure Ψ Ψ Ψ Ψ Ψ M,Ψ
$δ$ -neat Ψ Ψ Ψ Ψ Ψ M,Ψ
neat P,Ψ Ψ Ψ Ψ Ψ M,Ψ
c-real + +[SS,2.4] S[JM,7.3] S[JM,7.3] + M
p-real Sz[SS,2.8] + S S +[SS,2.4] M
$ψ$ -com P[BS,3.4] + + + M
$μ$ -com P,W[BS,3.5] W W[JM,7.2] + W[JM,7.2] M
$η$ -com
nearly real [JM,6.1] P,Sz Q[SS,2.8] S S + M
$λ$ -com P[BS,3.4],
W[BS,3.5] W W[BS,3.5] + [Rio,3.7] W[JM,7.2] +

References

[A] A. V. Archangel'skii, The Star Method, new classes ofspaces and countable compactness, Soviet Math Dokl, 21,7 (1980),550-554.
[BvD] R.L. Blair and E. van Douwen, Nearly RealcompactSpaces, Top. Appl. 47,3 (1992), 209-221.
[BLS] R.L. Blair and M.A. Swardson, Spaces with an OzStone-Cech Compactification, Top. Appl., 36 (1990), 73-92.
[BS] I.E. Blum and S. Swaminathan, Ideal Equivalences forAlmost Realcompact Spaces, Comm. Math. Univ. Carol., 21,1 (1980),81-95.
[Dy1] N. Dykes, Mappings and Realcompact Spaces, PacificJ. Math. 31 (1969), 347-358.
[Dy2] N. Dykes, Generalizations of Realcompact Spaces,Pacific J. Math. 33 (1970), 571-581.
[F] Z. Frolik, A generalization of realcompact spaces,Czech. Math. J., 13 (1963), 127-138.
[GP] R.J.Gardner and W.F.Pfeffer,Borel Measures, in"Handbook of Set-Theoretic Topology", Elsevier, 1984, 961-1043.
[GJ] L.Gillman and M.Jerison, "Rings of Continuous Functions",University Series in Higher Math, Van Nostrand, Princeton New Jersey, 1960.
[JM] D.Johnson and M.Mandelker, Functions withpseudocompact support, Gen. Top. Appl., 3 (1973), 331-338.
[Man] M.Mandelker,\ Supports of Continuous Functions,Trans. Amer. Math. Soc., 156 (1971), 73-83.
[Mil] E.S.Miller, Closed Preimages of CertainIsocompactness Properties, Top. Proc., 13 (1988), 107-123.
[Mro] S.Mrowka, On the union of Q-spaces, Bull.Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys., 6 (1958),365-368.
[Mys] A.Mysior, A union of realcompact spaces, Bull.Acad. Polon. Sci. Ser. Sci. Math., 29,3-4 (1981),169-172.
[PW] J.R.Porter and R.G.Woods, "Extensions and Absolutesof Hausdorff Spaces," Springer-Verlag, New York, Heidelberg,Berlin, 1988.
[RR] M.D.Rice and G.D.Reynolds, Weakly Borel-completetopological spaces, Fund. Math., 105 (1980), 179-185.
[Rio] D.Riordan, "Cozero-sets and function spacetopologies," Ph.D. Dissertation, Carleton Univ., Ottawa,Ontario, Canada, 1972.
[Rud] M.E.Rudin, A normal space $X$ forwhich $X ✗ I$ is not normal, Bull. Amer.Math. Soc. 77,2 (1971), p.246.
[Sak] M.Sakai, A new class of isocompact spaces andrelated results, Pacific J. Math., 122,1 (1986), 211-221.
[Shi] T. Shirota, A class of topological spaces, OsakaMath. J., 4 (1952), 23-40.
[Sim] P.Simon, A note on Rudin's example of Dowkerspace, Comm. Math. Univ. Carol., 12,4 (1971),825-833.
[Ste] L.A.Steen and J.A.Seeback, "Counterexamples inTopology," Springer, New York, 2nd ed., 1978.
[SS] M.A.Swardson and P.J.Szeptycki, When $X$ ^* is a $P'$ space, Canadian Bulletin, to appear.
[W] M. Weir, "Hewitt-Nachbin Spaces," North Holland Math.Studies, American Elsevier, New York, 1975.

Last modified December 16, 1999

	real	comp unif	hyper-	strongly	iso
real	+	+[GJ,15.20]	+	+	+
comp unif	U	+	+[Dy1,3.1]	+	+
hyper-	U,P	P[BLS,3.9]	+	+	+
strongly	U,P,D	P,D	D[BLS,3.9]	+	+
iso	M,My,Ψ, U,P,D	Ψ,D	Ψ,D	Ψ[BLS,3.9]	+
almost	M[W,16.12], My[My], D[PW,6U]	D	D		+
weakly Bc	M,My, Ψ,D	D,Ψ	D,Ψ	Ψ	+
a-realcompact closed complete [GP,8.12]	M,My,Ψ, D,R[Sim]	D,Ψ	D,Ψ	Ψ	+
pure	M,My,Ψ, D,R	D,Ψ	D,Ψ	Ψ	+
$δ$ -neat	M,My,Ψ, D,R	D	D		+
neat	M,My,Ψ, D,R,P	D,P,Ψ	D,Ψ	Ψ	+[Sak,2.6]
c-real	M,My,H, S,B,D	B,D	B,D	B	B[SS,2.8]
p-real	M,My,H,U S,B,D,Sz	B,D,Sz	B,D,Sz	B,Sz	Sz[SS,2.8],B
$ψ$ -com	M,My,H, P,D	P,D	D
$μ$ -com	M,My,H, P,W,D	P,D	D
$η$ -com nearly real [JM,6.1]	M,My,H, P,S,Q,U B,D,Sz	Q,B,D, Sz	Q,B,D, Sz	Q,B,Sz	Sz,B
$λ$ -com	H,P,W	P

	almost	weakly Bc	a-real	pure	$δ$ -neat	neat
real	+[F,Thm 10]	+[GP,14.11]	+	+	+	+
comp unif
hyper-	P	P
strongly	P	P
iso	Ψ,R,P	R,P	P	P
almost	+	+[RR,2.1]	+[Dy2,1.6]	+	+	+
weakly Bc	Ψ[RR,Ex3]	+	+[GP,6.13]	+	+	+
a-realcompact closed complete [GP,8.12]	Ψ[BS,3.2], R	R[Sim]	+	+	+	+
pure	R,Ψ	R		+	+	+
$δ$ -neat	R,Ψ	R			+	+
neat	P,R,Ψ	P,R	P	P[Sak,3.8]		+
c-real	H,S,B	H,S,B	H[BS,4.0], S[BS3.3], B	B	B	B
p-real	H,S,B, Sz	H,S,B, Sz	H,S,B, Sz	B,Sz	B,Sz	B,Sz
$ψ$ -com	H,P	H,P	H[BS,4.0], P[BS,3.4]
$μ$ -com	H,P,W	H,P,W	H,P, W[BS,3.5]
$η$ -com nearly real [JM,6.1]	H,P,S, Sz	H,P,S, Sz	H,P,S, Sz	Sz	Sz	Sz
$λ$ -com	H,P,W	H,P,W	H[BS,4.0], P[BS,3.4], W[BS,3.5]

	c-real	p-real	psi-com	mu-com	eta-com	lambda-com
real	+	+	+	+	+	+
comp unif		+			+
hyper-	P	+			+
strongly	P	+[SS,2.6]			+
iso	P,Ψ	Ψ	Ψ	Ψ	Ψ	M,Ψ
almost	+[Dy2,3.3]	+	+[BS,2.1]	+	+	M[BS,3.1]
weakly Bc	Ψ	Ψ	Ψ	Ψ	Ψ	M,Ψ
a-realcompact closed complete [GP,8.12]	Ψ	Ψ	Ψ	Ψ[BS,3.2]	Ψ[BS,3.2]	M, Ψ[BS,3.2]
pure	Ψ	Ψ	Ψ	Ψ	Ψ	M,Ψ
$δ$ -neat	Ψ	Ψ	Ψ	Ψ	Ψ	M,Ψ
neat	P,Ψ	Ψ	Ψ	Ψ	Ψ	M,Ψ
c-real	+	+[SS,2.4]	S[JM,7.3]	S[JM,7.3]	+	M
p-real	Sz[SS,2.8]	+	S	S	+[SS,2.4]	M
$ψ$ -com	P[BS,3.4]		+	+	+	M
$μ$ -com	P,W[BS,3.5]	W	W[JM,7.2]	+	W[JM,7.2]	M
$η$ -com nearly real [JM,6.1]	P,Sz	Q[SS,2.8]	S	S	+	M
$λ$ -com	P[BS,3.4], W[BS,3.5]	W	W[BS,3.5]	+ [Rio,3.7]	W[JM,7.2]	+