Mioduszewski, J.

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Institute of Mathematics AS CR, v. v. i.

Toposym 3Jerzy MioduszewskiOn a method which leads to extremally disconnected coversIn: Josef Novák (ed.): General Topology and its Relations to Modern Analysis and Algebra,Proceedings of the Third Prague Topological Symposium, 1971. Academia Publishing House of theCzechoslovak Academy of Sciences, Praha, 1972. pp. 309--311.Persistent URL: http://dml.cz/dmlcz/700722Terms of use:© Institute of Mathematics AS CR, 1972Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.cz309 ON A METHOD WHICH LEADS TO EXTREMALLY DISCONNECTED COVERS J. MIODUSZEWSKI Katowice Recently Hager [4] has found a simple method (using Kuratowski-Zorn Lemma) to get for each compact HausdorfF space the associated extremally disconnected cover (the construction due originally to Gleason [3] and simplified by Rainwater [8]). The aim of this note is to show that this method may be applied in order to get extremally disconnected covers for arbitrary HausdorfF spaces. Throughout this text all the spaces are assumed (this is in fact not always neces-sary) to be HausdorfF and all the maps are assumed to be continuous. A map p : Y—?+X will be said to be irreducible ifF for each closed subset F of Y we have CI p(F) =# X whenever F =f= Y; in the case of compact HausdorfF spaces this notion coincides with the usual one. Irreducible maps Y -» X will be said also to be covers of X9 and in the case when Y is extremally disconnected, extremally disconnected covers of X. It was shown by L. Rudolf and the author in [7] that \. If p :Y -* X is irreducible, then for each V regularly open in Y there exists a U regularly open in X9 such that V= Int p"\Cl U) = Int CI p~\U)9 and this determines U for a given V. The above lemma established a one-to-one correspondence, p#: U -* Int p~ *(C1 U)9 between the families of regularly open subsets of X and Y9 which preserves the Boolean operations. It is easy to check that if u is an ultrafilter in X (we consider only filters consisting of regularly open subsets), then p+u = {p*(U) : U e u} is ari ultrafilter in Y. Thus the underlying set of Ymay be regarded as a subset of the family R(X) of all ultrafilters in X9 and consequently, every class of non-homeomorphic covers of X is a set. Let Cov X be a fixed set of covers of X such that each cover of X is homeomorphic to one of Cov X. The following cancellation law may be found in [7] (see [8] for the compact case): 2. If p :Y -* X is irreducible and h ;Y -+Y is such that p © h *-=. p9 then h is the identity. 310 J. MЮDUSZEWSKI This leads to a corollary that we get a partial ordering in Cov X if we set p -< q for covers p : У-» X and q : Z -* X whenever there exists Һ:Z -* У such that p o Һ = q. Let iî, R cz R(X), be a set of ultŕafilters in X such that for each x in .Y there exists a ti in R such that x є П{C1 U: U є u}; in particular, the whole R(X) satisfies this condition. A cover p : У-* X will be said to be an R-cover if for each u in R there exists a j; in У such that y є П{C1 P*(U) : U є u}; then p(y) is the only point in П{C1 U : U є u}. Let Covк Z be the set of all R-covers of X. Let us note an easy fact (cf. [3]): 3. // V is regularly open in У, then the projection (Y — V) + Cl V -* У is aw R-cover for any R, R a Л(У); here ** + " stands for the free union. The projection is not a homeomorphism unless Vis closed-open. This leads to the conclusion that the maximal elements in CovRZ are extremally disconnected covers. To examine the set Covк X with respect to maximal elements, note first that 4. If p :Y-+ X and q : Z -* X are R-coveгs, h : Z ~>Y is such that p o Һ = q and uєR, then h(z) = y, whenever {y} = f|{Clp*(U) \Uєu} and {z} = П{C1 q*(U): :Uєu}. The proof consists in the calculation which makes use of Assertion 1, that ҺГ Қy) = ==- f|{Clq*(U) \Uєu} (cf. [7], p. 29, foг a similar calculation in a more general situation). Now we shall prove 5. The set CovÄ-Y řs closed with respect to inverse limits of directed systems. Proof. Let P be a directed system of JR-covers of X. Let Z be the inverse limit space of the system and let px : Z -* X be the projection. The projection px is onto. To see this, let x є X. Let u in -R be such that {x} = = П{C1 U : U є u}. If У is a space of the system P, then let zү be the only point in П{C1 (px)* (U) : U є u}, where Px:Y-+X is a map from the system. The point z whose coordinates are zү just defined, У running over the spaces of the system, is an element of Z, in virtue of Assertion 4. Clearly, px(z) = x. The projection px is irreducible. In fact, let F be closed in Z and such that F + Z. Consider W' -= Z — F, open and non-empty. Since the system P is diгected, there exists a space У in the system and a non-empty open subset V of У such that pÇ\V) c: Ж Thus we have Cl pү(F) cz У - V, and consequently Clpy(Ғ) Ф У. From the irreducibility of px it follows that Cl px(F) 4= X. The projection px is an K-cover. In fact, let u є R and let us take an x in X such that {x} = f){Cl 17 : U є u}. Let z in Z be taken for x and for u as at the begin-ning of the proof, so that px(z) = y. Foг this z we get z є f){Cl (p^)* (17): 1/ є u}. J. MIODUSZEWSKI 311 Now, using the Kuratowski-Zorn Lemma, we get the maximal elements in CovRX. Note that the maximal elements in CovK X are maximal in the whole Cov X if we neglect a difference in topologies modulo regularly open subsets (i.e., if we regard topologies as equivalent if the families of regularly open sets in these topologies are equal). To see this, let p : E -> X be a maximal element in CovK X. The space E is extremally disconnected, in virtue of the remark following Assertion 3. Then the announced fact follows from the assertion below (Blaszczyk [1]; for more usual version, see Gleason [3] and Flachsmeyer [2]): 6. If E is extremally disconnected then each irreducible map onto E is a bijec-tion which preserves regularly open subsets. There are many extremally disconnected covers for a given X9 e.g. depending on the choice of R9 R cz R(X). A partial ordering in the set E Cov X9 of extremally disconnected covers of X9 is introduced if we set p ^ q iff there exists an h such that qo h = p (notice the change of the order of composition in comparison to the ordering in the whole Cov X). The map h which realizes the inequality is an "embed-ding" onto a dense subset, if we neglect the difference of topologies modulo regularly open subsets. We get the greatest extremally disconnected cover of X9 if we take for R the whole R(X); this is the Iliadis extremally disconnected cover as defined in [7], p. 31 (see also [6]). References [1] A. Blaszczyk: A characterization of extremally disconnected spaces by means of irreducible maps. Notes in Mathematics, Silesian University, Katowice (to appear). [2] /. Flachsmeyer: Topologische Projektivraume. Math. Nachr. 26 (1963), 355—365. [3] A. M. Gleason: Projective topological spaces. Illinois J. Math. 2 (1958), 482—489. [4] A. W. Hager: The projective resolution of a compact space. Proc. Amer. Math. Soc. 28 (1971), 262-266. [5] C. HAuaduc: A6COJIK>TM xayczjopcfroBMx npocrpaHCTB. flAH CCCP 149 (1963), 22—25. [6] /. Mioduszewski andL. Rudolf: //-closed spaces and projectiveness. Contributions to Extension Theory of Topological Structures (Proc. Sympos., Berlin, 1967). Deutsch. Verlag Wissensch., Berlin, 1969, 153-156. [7] /. Mioduszewski and L. Rudolf: /f-closed and extremally disconnected Hausdorff spaces. Dissertationes Math. 66, 1969. [8] /. Rainwater: A note on projective resolutions. Proc. Amer. Math. Soc. 10 (1959), 734—735. SILESIAN UNIVERSITY, KATOWICE 

On a method which leads to extremally disconnected covers

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On a method which leads to extremally disconnected covers

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Institute of Mathematics AS CR, v. v. i.