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Metacompact space
Metacompact space
Metacompact space
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In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological space, there is a refinement that is again an open cover with the property that every point is contained only in finitely many sets of the refining cover.

A space is countably metacompact if every countable open cover has a point-finite open refinement.

Properties

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The following can be said about metacompactness in relation to other properties of topological spaces:

Covering dimension

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A topological space X is said to be of covering dimension n if every open cover of X has a point-finite open refinement such that no point of X is included in more than n + 1 sets in the refinement and if n is the minimum value for which this is true. If no such minimal n exists, the space is said to be of infinite covering dimension.

See also

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References

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Metacompact space

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A metacompact space is a topological space in which every open cover admits a point-finite open refinement. Metacompactness is a refinement property in general topology that lies strictly between countable metacompactness and paracompactness, where the latter requires locally finite refinements of open covers. Every paracompact space is metacompact, but the converse fails; a classic counterexample is the Dieudonné plank, which is metacompact (and even hereditarily metacompact) yet not paracompact. Similarly, the Moore plane (also known as the Niemytzki plane) exemplifies a space that is countably metacompact but not metacompact. Key properties include the fact that every closed subspace of a metacompact space is metacompact, the product of a compact space and a metacompact space is metacompact, and a continuous closed surjection from a metacompact space onto another space preserves metacompactness. A notable characterization is Watson's theorem, which states that every pseudocompact metacompact space is compact. Examples of metacompact spaces include compact spaces, Lindelöf spaces that are countably metacompact, and the real line with the discrete topology.

Definition and Basic Concepts

Definition

In topology, a topological space XX is defined to be metacompact if every open cover of XX admits an open refinement that is point-finite. An open cover of XX is a collection U={Ui∣i∈I}\mathcal{U} = \{U_i \mid i \in I\} of open subsets of XX such that their union equals XX, i.e., X=⋃i∈IUiX = \bigcup_{i \in I} U_i. A refinement of U\mathcal{U} is another open cover V\mathcal{V} such that for each V∈VV \in \mathcal{V}, there exists some U∈UU \in \mathcal{U} with V⊆UV \subseteq U. The key condition of point-finiteness requires that for every point x∈Xx \in X, xx belongs to only finitely many members of V\mathcal{V}. To illustrate a non-point-finite cover, consider the real line R\mathbb{R} with the standard topology and the point x=0x = 0; the collection {(−1/n,1/n)∣n=1,2,3,… }\{ (-1/n, 1/n) \mid n = 1, 2, 3, \dots \} is an open cover of the interval (−1,1)(-1, 1), but it is not point-finite at 00 since 00 lies in infinitely many of these nested open intervals. Metacompactness is a weaker property than paracompactness, where the refinement is required to be locally finite rather than merely point-finite.

Countable Metacompactness

A topological space is countably metacompact if every countable open cover has a point-finite open refinement. This means that for any countable collection of open sets covering the space, there exists a refinement consisting of open sets such that each point in the space belongs to only finitely many sets of the refinement. In contrast to the general definition of metacompactness, which requires every open cover—regardless of cardinality—to admit such a point-finite open refinement, countable metacompactness imposes the condition only on countable covers. This restriction makes countable metacompactness a strictly weaker property than metacompactness, as a space may refine point-finitely all countable open covers while failing to do so for some uncountable open cover, where points might require membership in infinitely many sets of any attempted refinement. For instance, the combination of countable metacompactness with the Lindelöf property (where every open cover has a countable subcover) is necessary to imply full metacompactness, underscoring that countable metacompactness alone does not suffice. Thus, there exist topological spaces that are countably metacompact but not metacompact, highlighting the distinction between these notions in general topology.

Properties and Relations

Key Properties

A metacompact space is orthocompact. An orthocompact space is a topological space in which every open cover has an interior-preserving open refinement, meaning that for any open cover U\mathcal{U}, there exists an open refinement V\mathcal{V} such that the interior of the union of any subcollection of V\mathcal{V} equals the union of the interiors of its members. The proof sketch relies on the point-finite refinement guaranteed by metacompactness: given an open cover U\mathcal{U}, select a point-finite open refinement V\mathcal{V}; since each point lies in only finitely many members of V\mathcal{V}, the intersections of subcollections are open, ensuring the refinement is interior-preserving. Every metacompact normal space is a shrinking space. A shrinking space is a topological space where every open cover U\mathcal{U} admits an open refinement V\mathcal{V} that shrinks with respect to U\mathcal{U}, meaning for each V∈VV \in \mathcal{V}, there exists U∈UU \in \mathcal{U} such that V‾⊂U\overline{V} \subset U. The normality of the space plays a crucial role in this implication, as it allows the construction of closed sets with disjoint interiors to separate points and ensure the closures of the refinement members are contained within the original cover elements, leveraging the point-finite refinement from metacompactness. This result, originally due to Lefschetz, highlights how the combination of metacompactness and normality strengthens covering properties, though coverage in some sources may lag behind post-1980s developments such as those in product space analyses. The product of a compact space and a metacompact space is metacompact. This follows from an application of the tube lemma, which states that if KK is compact and XX is any space, then for any open set WW containing K×{y}K \times \{y\} in K×XK \times X, there exists an open neighborhood UU of yy in XX such that K×U⊂WK \times U \subset W. To prove metacompactness of K×XK \times X where XX is metacompact, consider an arbitrary open cover U\mathcal{U} of K×XK \times X; by the tube lemma, for each fixed y∈Xy \in X, the slice {y}×K\{y\} \times K has a tube K×UyK \times U_y covered by finitely many elements of U\mathcal{U}; since XX is metacompact, the collection {Uy}\{U_y\} has a point-finite open refinement V\mathcal{V}, and pulling back via the tube lemma yields a point-finite open refinement of U\mathcal{U} for K×XK \times X.

Relations to Other Notions

A paracompact space is defined as a topological space in which every open cover admits a locally finite open refinement. Every paracompact space is therefore metacompact, since a locally finite open cover is a special case of a point-finite open cover, where each point belongs to only finitely many sets in the refinement. However, the converse does not hold; there exist metacompact spaces that are not paracompact, such as the Dieudonné plank. Compact spaces are metacompact as a consequence of being paracompact: compactness ensures that every open cover has a finite subcover, which is locally finite and thus point-finite. Similarly, all metric spaces are paracompact and hence metacompact; this includes not only basic Euclidean spaces but also more advanced examples like separable Hilbert spaces and Fréchet spaces, which admit partitions of unity and satisfy the necessary covering properties. In summary, the implications form a strict hierarchy: compactness implies paracompactness, which in turn implies metacompactness, though each step is irreversible in general topology.

Examples and Counterexamples

Dieudonné Plank

The Dieudonné plank is a topological space introduced by Jean Dieudonné in 1944 as a key counterexample in the study of separation axioms and covering properties in general topology. It serves to illustrate the distinction between metacompactness and paracompactness, being an example of a normal metacompact space that fails to be paracompact.

Construction

The Dieudonné plank is constructed as the space [0,ω1]×[0,ω][0, \omega_1] \times [0, \omega], where ω1\omega_1 denotes the first uncountable ordinal and ω\omega the first infinite ordinal, equipped with the order topology on each factor and the product topology. This space can equivalently be viewed as the product of a discretized ordinal space on ω1+1\omega_1 + 1 (where all points except ω1\omega_1 are isolated, and neighborhoods of ω1\omega_1 are cocountable) and the ordinal space [0,ω]+1[0, \omega] + 1 with its order topology (where points are isolated except for neighborhoods of ω+1\omega + 1 being cofinite). The deleted Dieudonné plank, obtained by excluding the corner point (ω1,ω)(\omega_1, \omega), is non-normal but still metacompact, highlighting separation issues near the corner.

Proof of Metacompactness

The Dieudonné plank is metacompact because every open cover admits a point-finite open refinement, a property arising from the ordinal structure and the Noetherian base of sub-infinite rank possessed by the space. Specifically, the factor spaces each have Noetherian bases of rank 1, and their product inherits a Noetherian base of sub-infinite rank, which guarantees hereditary metacompactness via the construction of point-finite refinements from such bases. This refinement ensures that each point lies in only finitely many sets of the cover, leveraging the well-ordered nature of the ordinals to control intersections.

Proof of Non-Paracompactness

The Dieudonné plank is not paracompact, as there exists an open cover without a locally finite open refinement, particularly one involving bounded open intervals that cannot be refined locally finitely due to the uncountable cofinality at ω1\omega_1. In the original construction, considering a cover by non-trivial bounded open intervals leads to an absurdity in attempting a locally finite refinement, as neighborhoods near higher ordinals would require infinitely many overlapping sets without finite local support. Furthermore, the space lacks a totally ordered local base at the corner point (ω1,ω)(\omega_1, \omega), preventing a base of finite rank and thus failing paracompactness, while the deleted version highlights non-normality issues tied to the excluded point's role in separation.

Moore Plane

The Moore plane, also known as the Niemytzki plane or tangent disc space, is constructed as the upper half-plane Γ=R×[0,∞)\Gamma = \mathbb{R} \times [0, \infty) equipped with a specific topology. Points in the interior P={(x,y)∣y>0}P = \{(x, y) \mid y > 0\} have the standard Euclidean neighborhood basis consisting of open discs Bd((x,y),r)B_d((x,y), r) for 0<r<y0 < r < y. For points on the boundary line L=R×{0}L = \mathbb{R} \times \{0\}, the neighborhood basis consists of sets of the form {(s,0)}∪Bd((s,r),r)\{(s, 0)\} \cup B_d((s, r), r) for r>0r > 0, where the disc is tangent to the x-axis at (s,0)(s, 0). This space serves as a key example of a countably metacompact space that fails to be metacompact. For any countable open cover, there exists a point-finite open refinement. However, there exists an open cover without a point-finite open refinement. Additionally, the Moore plane is completely regular but not normal.

Theorems and Characterizations

Watson's Theorem

In 1981, W. Stephen Watson proved that any pseudocompact metacompact space is compact, providing a characterization of compactness in Tychonoff spaces via the combination of metacompactness and pseudocompactness (where pseudocompactness means that every continuous real-valued function on the space is bounded). The converse direction—that every compact Tychonoff space is both metacompact and pseudocompact—is a standard result in topology, as compactness implies these properties directly. This theorem resolves an open question posed by Aull and others in the 1970s, generalizing earlier results such as the compactness of countably compact metacompact spaces (due to Arens and Dugundji in 1950) and the compactness of pseudocompact paracompact spaces. Watson's result was published in the Proceedings of the American Mathematical Society and independently obtained around the same time by P. H. Scott and H. Förster, highlighting its significance in advancing characterizations of compactness beyond paracompactness. The theorem applies specifically to completely regular spaces (including Tychonoff spaces), and post-1981 literature has not identified significant extensions to non-Tychonoff spaces, though related results explore variations like metacompactness in other compactness-like properties.

Proof Outline

To prove sufficiency, let XX be a pseudocompact metacompact space and θ\theta an arbitrary open cover of XX. Since XX is metacompact, there exists a point-finite open refinement U\mathscr{U} of θ\theta; by regularity of XX, assume U‾⊆V\overline{U} \subseteq V for some V∈θV \in \theta whenever U∈UU \in \mathscr{U}. Pseudocompactness implies XX is Baire, so there is a π\pi-base B\mathscr{B} for XX such that for each B∈BB \in \mathscr{B} and U∈UU \in \mathscr{U}, either B⊆UB \subseteq U or B∩U=∅B \cap U = \emptyset. Inductively construct a sequence {An}\{A_n\} in B\mathscr{B} where each AnA_n is disjoint from the closure of the union of all U∈UU \in \mathscr{U} that intersect some previous AmA_m for m<nm < n, and AnA_n intersects some U∈UU \in \mathscr{U} not yet "accounted for." If this sequence is infinite, pseudocompactness (which forbids infinite discrete collections of open sets) yields a limit point a∈Xa \in X, but aa must lie in some U(a)∈UU(a) \in \mathscr{U} intersecting infinitely many AnA_n, contradicting their disjointness properties. Thus, the sequence is finite, say up to n=kn = k, and since U\mathscr{U} is point-finite, only finitely many elements of U\mathscr{U} intersect the AmA_m for m<km < k. These finitely many elements of U\mathscr{U} (and hence of θ\theta) cover a dense subset of XX, and by regularity, they cover all of XX, proving every open cover has a finite subcover.

Covering Dimension Connection

In , the covering dimension of a topological space XX, denoted dim⁡X\dim X, is defined as the smallest non-negative integer nn such that every open cover of XX admits a point-finite open refinement in which no point of XX belongs to more than n+1n+1 sets of the refinement; if no such finite nn exists, then dim⁡X=∞\dim X = \infty. This definition captures the intuitive notion of dimension by controlling the local overlapping of sets in refinements of covers, ensuring that the "thickness" of the space is bounded in a precise manner. Metacompactness plays a foundational role in this context by guaranteeing the existence of point-finite open refinements for every open cover, which serves as a necessary condition for XX to have finite covering dimension, since such dimensions require refinements where the multiplicity at each point is finite (and in fact bounded). However, metacompactness alone does not impose a uniform bound on the multiplicity across the space, allowing metacompact spaces to exhibit infinite covering dimension, as the point-finiteness is not sufficient to restrict overlaps to a fixed finite number. In paracompact Hausdorff spaces, a stronger condition known as nn-bounded metacompactness—where refinements have order at most nn—is equivalent to having covering dimension n−1n-1, highlighting how bounded control over point-finiteness aligns directly with finite-dimensionality.

References

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