Fan-Gottesman Compactification and Scattered Spaces

Ceren Sultan Elmali 1  and Tamer Ugŭr 1
  • 1 Department of Mathematics, Faculty of Science, 25240, Erzurum, Turkey
Ceren Sultan Elmali
  • Corresponding author
  • Department of Mathematics, Faculty of Science, Ataturk University, Erzurum, 25240, Turkey
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
and Tamer Ugŭr

Abstract

Compactification is the process or result of making a topological space into a compact space. An embedding of a topological space X as a dense subset of a compact space is called a compactification of X. There are a lot of compactification methods but we study with Fan- Gottesman compactification. A topological space X is said to be scattered if every nonempty subset S of X contains at least one point which is isolated in S. Compact scattered spaces are important for analysis and topology. In this paper, we investigate the relation between the Fan-Gottesman compactification of T3 space and scattered spaces. We show under which conditions the Fan-Gottesman compactification X* is a scattered.

1 introduction and some Preliminaries

The notion of compactness or compactification is an important concept in general topology as well as in a branch of mathematics and quantum physic. A compactification of a space X is a compact space containing X as a dense subspace. There are a lot of compactification methods applying different topological spaces such as Aleksandrov (one-point), Wallman, Stone-Cech. In 1929, Aleksandrov proved that all locally compact Hausdorff spaces may be completed to a compact hausdorff by the addition of one point [1]. In 1937, Cech constructed the compactification of a space by using the diagonal product of all continuous functions [3] while Stone was using Boolean algebras and rings of continuous functions [12] to do this. Henry Wallman introduced compactification of spaces having a normal base which is also called Wallman compactification [13]. Recall the Fan-Gottesman compactification is defined by Ky Fan and Noel Gottesman.

In 1952, Let β be a class of open sets in X. If it satisfies the following three conditions, it is called a normal base.

  1. β is closed under finite intersections
  2. If Bβ, then X − clX Bβ, where clX B denotes closure of B in X.
  3. For every open set U in X and every Bβ such that clX BU, there exists a set Dβ such that clX BDclX DU.

We consider a regular space having a normal base for open sets i.e., which satisfies the above three properties of normal base A chain family on β is a non-empty family of sets of β such that

clXB1clXB2clXBn
for any finite number of sets Bi of the family. Every chain family on β is contained in at least one maximal chain family on β by Zorn’s lemma. Maximal chain families on β will be denoted by letters as a*, b*,..., and also the set of all maximal chain families on β will be denoted by (X, β) *. (X, β) * is a compact Hausdorff spaces and is a compactification of our regular space. Whose topology is defined as follow. For each Bβ, let
τ(B)={b*(X,β)*:Bb*}
Then the topology of (X, β) * is defined by taking
β*={τ(B):Bβ}
as a base of open sets. Afterwards this compactification is called Fan-Gottesman compactification [7] .

Now, we defined this compactification via ultra-open filter in [5].

Definition 1.1

Let X be a T3 space and FX the subcollection of all open ultrafilters on X. For each open set OX, define O*FX to be the set

O*={G^FX:VckXVO,OisopeninXandV,OG^}
Let Φ is the {O* : O is open subset of X} set. It is clear that Φ is the base for open sets of topology on FX. FX is a compact space and the Fan-Gottesman compactifications of X.

In order to avoid the confusion between FX and (X, β)*, we will use X* when it regarded as Fan-Gottesman compactification of X.

On the other hand, for each closed set DX, we define D*X* by

D*={G^X*:GDforsomeGinG^}.

The following properties of X* are useful

  • If UX is open, then X*U* = (X*U)*
  • If DX is closed, then X*D* = (X*D)*
  • If U1 and U2 are open in X, then (U1U2)*=U1*U2* and (U1U2)*=U1*U2*

Properties We consider the map fX : XX* be defined by fX (x) =Ĝx, the open ultrafilter converging to x in X. Then the following properties hold.

  1. If U is open in X, then fX (Ū) = U*. In particular fX (X) is dense in X*.
  2. fX is continuous and it is an embedding of X in X* if and only if X is a T3-space.
  3. If U1 and U2 are open subsets of X, then fX (Ū1U2) = fX (Ū1) ∩ fX (Ū2).
  4. X* is a compact T2-space.

Recal that X*X is called as Fan-Gottesman remainder and we get following theorems in.

Theorem 1.1

Let X be a topological space. Suppose that X*X contains an infinite subspace having the topology of finite complements. Then there exists a sequence H1, H2,... of closed noncompact subsets of X which are pairwise disjoint.

Theorem 1.2

Let X be a topological space. Then the following statements are equivalent:

  1. The Fan-Gotteman compactification of X is its weak reflection in compact spaces.
  2. The space X has a weak reflection in compact spaces.
  3. There exists such that any pairwise disjoint family of closed sets in X contains at most k noncompact elements.
  4. Every infinite sequences F1, F2,... of closed sets such that FpFq is compact for pq has a compact member
  5. The Fan-Gottesman remainder of X is finite.

A topological space X is said to be scattered if every non-empty subset S of X contains at least one point which is isolated in S. Compact scattered spaces are very important tools for analysis and topology. In 1972, Mrowka and others characterized compact scattered space [11]. In 1974, Kannan and Rajagopalan showed that a 0-dimensional, Lindolef, scattered first countable Hausdorff Space admits a scattered compactification [9]. In 1979, Moran characterize the class of scattered compact orders among the compact orders by means of properties of their continuous real valued functions [10]. In 2007, M.Henriksen and others established a new generalization of scattered spaces called SP-scattered [8]. In 2016, Al-Hajri and others characterized spaces such that their Aleksandrov (one-poin) compactification is a scattered space. They presented image classificaiton problem as an application of scattered space [2]. In 2018, Cioban and Budanaev present some geometrical and topological concepts to accommodate the needs of information theories. They established that Khalimsky topology on the discrete line is unique as the minimal symmetric digital topology on ℤ [4].

Definition 1.2

Let X be a topological space and SX.S is called a scattered subset of X if S considered as a topological space is scattered. Also if clS is scattered, then S is scattered.

Proposition 1.1

Let X be a TO-topological space and S be a subset of X. clS is scattered if and only if S and clS are scattered.

Proposition 1.2

Let X be a TO-topological space and S1, S2 be two scattered subsets of X. The following statements hold:

  1. The intersection of Sl and S2 is a scattered subset of X.
  2. The union of Sl and S2 is a scattered subset of X.

Corollary 1.3

Let X be a topological space and {Si : iI} be a finite collection of scattered subsets of X. Then iISiSi is a scattered subset of X.

The following remarks are very important for topologists. Because the results are scattered space definition and the construction of compactification.

Remark 1.4

Let K(X) be a compactification of a topological space X. If K(X) is a scattered then X is also a scattered space.

Remark 1.5

Let X be a scattered topological space and SX. Then S is also a scattered space.

2 Main Results

Theorem 2.1

Let X* be Fan-Gottesman compactification of X, T3-space. Then X* is a scattered if and only if X and X*X are scattered.

Proof

(⇒) It is obviously that if X* is a scattered then X and X*X are scattered due to hereditary properties of scattered space.

(⇐) Assume that X and X*X are scattered and let S be a subset of X*. We think about two cases:

Case 1: SX or SX*. So there exists an element x of S and an open set U of X such that US = {x}. Case 2: SX ≠ 0 and SX* ≠ 0. Since X is a scattered and SX ≠ 0. There are an element z of SX and an open set U of X such that U ∩ (SX) = {x}. Assume that (US) ∩ (X*X) ≠ 0. Because X*X is scattered there exists an element z of (US) ∩ (X*X) and an open set O of X such that O ∩ [(US) ∩ (X*X)] = {z}. Therefore either SO = {z} or SO = {x, z}. It is clear that SO = {z}, then z is an isolated point of S. Assume that SO = {x, z}, since X* is Hausdorff space and zX*X, there is an open set V of X* such that xV and zV. Therefore (VUO) ∩ S = {x}. Thus S is scattered.

Theorem 2.2

Let X be a T3-space such that its Fan-Gottesman compactification remainder is finite. Then the Fan-Gottesman compactifiction X* is a scattered if and only if X is a scattered space.

Proof

We get (⇒) It is obviously that if the Fan-Gottesman compactification X* is a scattered then X is a scattered space due to hereditary properties of scattered space.

(⇐) Since X*X is finite and X* is Hausdorff space, X*X is a discrete space. Then X*X is a scattered. It is clear that X* is a scattered space from Theorem 2.1.

Theorem 2.3

Let X be a T3-space such that its Fan-Gottesman compactification is a scattered space. If is a collection of disjoint non-compact, closed sets of X, then there exists an open set U of X and F such that

  1. There exists a closed, non-compact set K of X such that KFU
  2. For each non-compact, closed set DT with T − {F}, DU.

Proof

Let A = {ξX* : ∃T, Tξ}. Since X* is a scattered, there exists ξA such that ξ is an isolated point in A. Therefore there exists an open set U of X such that U*S = {ξ} Thus ξU*, so that there exists Kξ such that KU. Since ξS, let F such that Fξ. Then KFU.

Let T − {F} and D be a non-compact, closed set of X such that DT. Then there exists 𝒟S such that D𝒟. It is clear that 𝒟ξ for elements of T are disjoints; so that 𝒟U*. Hence DU.

References

  • [1]

    Aleksandrov P.S and Urysohn P., Memorie sur les espaces topologiques compacts. Koninkl. Nederl. Akad. Wetensch., Amsterdam (1929)

  • [2]

    Al-Hajrh M., Belaid K. and Belaid L., Scattered Spaces,Compactifications and An Application to Image Classification Problem, Tatra Mt. Math. Publ., 66(2016) 1–12.

  • [3]

    Cech E., On bicompact spaces Ann.of Math.(2) 38:4(1937) 823–844

  • [4]

    Cioban M. and Budanaev I., Scattered and Digital Topologies in Image Processing, Proceeding of the Conference on Mathematical Foundations of Informatics, 2–6 July 2018, Chisinau, Republic of Moldova.

  • [5]

    Elmalh C. and Ugur T., FG-morphisms and FG-extensions, Hacettepe Journal of Mathematics and Statistics, 43 (6) (2014), 915–922.

  • [6]

    Elmalh C., Ugur T. and Kopuzlu A., Weak Refiections and Remainders in Compactifications, Palestine Journal of Mathematics, 3(2) (2014), 289–292.

  • [7]

    Fan K and Gottesman N., On compactification of freudenthal and Wallman. Indag. Math. 13 (1952), 184–192.

  • [8]

    Henriksen M., Raphael R. and Woods R.G, Sp-Scattered Space; a new generalization of scattered space, Comment. Math. Univ. Carolin. 48(3) (2007), 487–505.

  • [9]

    Kannan V. and Rajagopalan M., On scattered Spaces, Proceedings of the American Mathematical Society 43(2) (1974), 402–408.

  • [10]

    Moran, G. On scattered Compact Ordered Sets, Proceedings of the American Mathematical Society 75(2) (1979), 355–360.

  • [11]

    Mrowka S., Rajagopalan M. and Soundararajan T., A characterization of compact scattered spaces through chain limits (Chain compact spaces), Proc. Pittsburgh Conference in Topology II (1972), Academic Press, New York.

  • [12]

    Stone M.H., Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc.41 (1937) 375–481

  • [13]

    Wallman H., Lattices and Topological Spaces. Annals of Mathematics 39, (1938) 112–126

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Aleksandrov P.S and Urysohn P., Memorie sur les espaces topologiques compacts. Koninkl. Nederl. Akad. Wetensch., Amsterdam (1929)

  • [2]

    Al-Hajrh M., Belaid K. and Belaid L., Scattered Spaces,Compactifications and An Application to Image Classification Problem, Tatra Mt. Math. Publ., 66(2016) 1–12.

  • [3]

    Cech E., On bicompact spaces Ann.of Math.(2) 38:4(1937) 823–844

  • [4]

    Cioban M. and Budanaev I., Scattered and Digital Topologies in Image Processing, Proceeding of the Conference on Mathematical Foundations of Informatics, 2–6 July 2018, Chisinau, Republic of Moldova.

  • [5]

    Elmalh C. and Ugur T., FG-morphisms and FG-extensions, Hacettepe Journal of Mathematics and Statistics, 43 (6) (2014), 915–922.

  • [6]

    Elmalh C., Ugur T. and Kopuzlu A., Weak Refiections and Remainders in Compactifications, Palestine Journal of Mathematics, 3(2) (2014), 289–292.

  • [7]

    Fan K and Gottesman N., On compactification of freudenthal and Wallman. Indag. Math. 13 (1952), 184–192.

  • [8]

    Henriksen M., Raphael R. and Woods R.G, Sp-Scattered Space; a new generalization of scattered space, Comment. Math. Univ. Carolin. 48(3) (2007), 487–505.

  • [9]

    Kannan V. and Rajagopalan M., On scattered Spaces, Proceedings of the American Mathematical Society 43(2) (1974), 402–408.

  • [10]

    Moran, G. On scattered Compact Ordered Sets, Proceedings of the American Mathematical Society 75(2) (1979), 355–360.

  • [11]

    Mrowka S., Rajagopalan M. and Soundararajan T., A characterization of compact scattered spaces through chain limits (Chain compact spaces), Proc. Pittsburgh Conference in Topology II (1972), Academic Press, New York.

  • [12]

    Stone M.H., Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc.41 (1937) 375–481

  • [13]

    Wallman H., Lattices and Topological Spaces. Annals of Mathematics 39, (1938) 112–126

OPEN ACCESS

Journal + Issues

Search