Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

A SURVEY OF J-SPACES

E. MICHAEL

Abstract. This note is a survey of J-spaces.

1. Basic concepts

A space X is a J-space if, whenever {A, B} is a closed cover of X with A ∩ B compact, then A or B is compact. A space X is a strong J-space if every compact K ⊂ X is contained in a compact L ⊂ X with X\L connected. [As in [4], all maps are continuous and all spaces are Hausdorff.]

1.1. Every strong J-space X is a J-space. The two concepts coincide when X is locally connected, but in general (even for closed subsets of ℝ^{2}) they do not.

2. Examples

2.1. A topological linear space X is a (strong) J-space if and only if X ≠ ℝ.

2.2. If X and Y are connected and non-compact, then X × Y is a strong J-space.^{1}

2.3. Let Y be a compact manifold with boundary B, and let A ⊂ B. Then Y \A is a (strong) J-space if and only if A is connected.

2000 Mathematics Subject Classification. Primary 54D20; Secondary 54D30 54D45 54E45 54F65.

Key words and phrases. J-spaces, covering properties. The author was an invited speaker at the Ninth Prague Topological Symposium. This contribution is excerpted from a published article. Reprinted from Topology and

its Applications, Volume 102, number 3, E. Michael, J-Spaces, pp. 315–339, Copyright (2000), with permission from Elsevier Science [4].

E. Michael, A survey of J-spaces, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 191–193, Topology Atlas, Toronto, 2002;

arXiv:math.GN/0204134.

1This was proved in [5].

3. Characterizations by closed maps

A map f : X → Y is called boundary-perfect if f is closed and bdry f^{−1}(y) is compact for every y ∈ Y. It follows from [3] that every closed map f : X → Y from a paracompact space X to a q-space Y is boundaryperfect.^{2}

3.1. A space X is a J-space if and only if every boundary-perfect map f : X → Y onto a non-compact space Y is perfect.

3.2. If X is a J-space, then every boundary-perfect map f : X → Y has at most one non-compact fiber. The converse holds if X is locally compact.

3.3. Let X be paracompact and locally compact. Then the following are equivalent.

(a) X is a J-space.

(b) Every closed map f : X → Y onto a non-compact, locally compact space Y is perfect.

(c) Every closed map f : X → Y onto a locally compact space Y has at most one non-compact fiber.

3.4. Let X be metrizable. Then the following are equivalent (a) X is a J-space.

(b) Every closed map f : X → Y onto a non-compact, metrizable space Y is perfect.

4. Characterization by compactifications

Call a set A ⊂ Y a boundary set for Y if Int A = ∅ and, whenever U ⊃ A is open in Y and {W_{1}, W_{2}} is a disjoint, relatively open cover of U\A, then no y ∈ A lies in \overline{W}_{1} ∩ \overline{W}_{2}. Call a set A ⊂ Y a strong boundary set for Y if Int A = ∅ and, whenever U ⊃ A is open in Y, then every y ∈ A has an open neighborhood V ⊂ U with V \A connected.

It is easy to see that, if Y is a manifold with boundary B, then every A ⊂ B is a strong boundary set for Y. And it follows from the proof of [1, Lemma 4] (or from [2, Proposition 3.5]) that, if Y is completely regular, then βX\X is a boundary set for βX.

4.1. Let Y be a compactification of X, and suppose either that X is locally compact or that Y is metrizable. Then the following are equivalent.

(a) X is a (strong) J-space.

(b) Y \X is connected and a (strong) boundary set for Y.

5. Preservation

5.1. J-spaces are preserved by boundary-perfect images. (False for strong J-spaces, even with perfect images.)

2q-spaces (see [3]) include all locally compact and all metrizable spaces.

A SURVEY OF J-SPACES 193

5.2. J-spaces and strong J-spaces are preserved by monotone, perfect preimages.

5.3. If X_{1}, X_{2} are connected, then X_{1} × X_{2} is a (strong) J-space if and only if either X_{1}, X_{2} are both (strong) J-spaces or both are non-compact.

5.4. Let {X_{1}, X_{2}} be a closed cover of X with X_{1} ∩ X_{2} compact. Then X is a (strong) J-space if and only if X_{1}, X_{2} are both (strong) J-spaces and X_{1} or X_{2} is compact.

5.5. If X is a (strong) J-space, so is every component of X. (False for J-spaces).

References

[1] Melvin Henriksen and J. R. Isbell, Local connectedness in the Stone-Čech compactification, Illinois J. Math. 1 (1957), 574–582. MR 20 #2688

[2] E. Michael, Cuts, Acta Math. 111 (1964), 1–36. MR 29 #5233

[3] , A note on closed maps and compact sets, Israel J. Math. 2 (1964), 173–176. MR 31 #1659

[4] , J-spaces, Topology Appl. 102 (2000), no. 3, 315–339. MR 2002b:54019

[5] Krzysztof Nowiński, Closed mappings and the Freudenthal compactification, Fund. Math. 76 (1972), no. 1, 71–83. MR 48 #2978

University of Washington, Seattle, WA, U.S.A.