Rev. Acad. Canar. Cienc., XVI (Núms. 1-2), 57-69 (2004) (publicado en julio de 2005)
ON SOME TOPOLOGICAL CONCEPTS VIA
THE LATTICE OF CLOSED SETS
M.N. Mukherjee & A. Sengupta*'
Abstract : The paper is devoted chiefly to the study of t he interplay between certain
topological concepts of a topological space X and sorne lattice concepts of L(X)-the
lattice of closed subsets of X under set inclusion . Among other things, the notions
of connectedness of a topological space and homeomorphism between two spaces
have been characterized in the language of lattices. Moreover, we introduce sorne
separation-like properties of a bounded distributive lattice, and investigate the interrelations
between these properties applied to L(X) and sorne separation axioms like
Hausdorffness and normali ty of a topological space X. Lastly, an equivalent description
of the notion of dimension of a topological space X is achieved in terms of a
lattice theoretic concept, introduced suita bly.
Key Words : Lattice of closed et , lattice of compact sets, principal ideal, prime
filter, depth of a lattice
2000 AMS Classification Code : 54H99, 54D30, 54D10, 54D15
1. lNTRODUCTION
It is well known that the lattice theoretic concepts have many interesting applications
to the development of various topics of general topology. A comprehensive
description of sorne such interplay of t he lattice-concepts vis-a-vis topological notions
can be had in the classical book of Birkhoff [l]. It is also seen from literature that
good many mathematicians have contributed a lot in this regard. For instance, Harding
and Pogel in a rather recent paper [3] have settled an open problem by proving
1*The second author acknowledges the financia! support from the University Grants C.S.I.R, New
Delhi for this work.
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that every lattice with 1 and O is embeddable in the lattice of topologies of sorne set
by an embedding preserving the 1 and O. In another recent paper [5], the authors
have studied the problem of reconstruction of topological spaces and continuous maps
from the lattice of open sets and the lattice homomorphisms respectively.
The intent of this paper is to endeavour to making sorne investigation along the
above trend of study. Our concern is mainly to unfold sorne interplay between certain
topological concepts and a lattice, the lattice of closed subsets of a topological space
X under set-inclusion.
In Section 2 we exploit the idea of principal ideals of L(X) to introduce a concept
of disconnectedness for a lattice, and this enables us to characterize the topological
connectedness of a space in terms of the concept, so defined. We ultimately prove that
the homeomorphism between two T1 topological spaces X and Y can be characterized
by lattice isomorphism between L(X) and L(Y). Another version of the last stated
result is obtained again in Section 3, where L(X) and L(Y) are replaced respectively
by K(X) and K(Y), where K(X) stands for the lattice of all compact subsets of a
compact, Hausdorff space X. Incidentally we see that there exists a covariant functor
from the category of Hausdorff spaces and continuous maps, to that of distributive
lattices and lattice homomorphisms.
Section 4 deals with sorne separation axioms of a topological space X with reference
to certain separation properties introduced suitably for a bounded distributive
lattice and applied to the lattice L(X) in particular.
The notion of dimension of a topological space X is known to be defined in terms
of irreducible closed subsets of X (see [7] for details). Our aim in the last section is
to give an alternative formulation of such a concept in terms of lattice language, and
this necessitates us to introduce a concept, termed depth of a lattice.
In what follows in this article, by a lattice (L , :S) we shall mean, unless otherwise
is stated explicitly, a bounded distributive latti ce with 1 and O as its largest and
smallest members respectively. Also, by a space X will be meant a topological space
X.
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2. LATTICE OF CLOSED SETS OF A TOPOLOGICAL
SPACE
As stated already in the introduction, our concern in this section is to associate to
each topological space X, a bounded distributive lattice L(X), the lattice of closed
subsets of X under set inclution, and study sorne interplay between X and L(X).
For doing this, we first append sorne standard notions of lattice theory as follows.
Definition 2.1 [1,6] : Suppose (L, :S) is a lattice. A nonempty subset I of L is
called an ideal of L, if
(i) a, b E L ==;.. a V b E l and (ii) a E l and b E L with b :S a ==;.. b E /.
We observe that for each x E L, I(x) = {y E L : y :S x} is the smallest ideal
containing x; we call /(x) the principal ideal of L generated by x [1 ,6]. Obviously,
/(O) = {O} and / (1) = L. If x tJ_ {O, l} then I(x) is called a non-t rivial principal
ideal. We also see that {O,x} ~ I(x), for each x E L. If I(x) = {O,x}, then we say
that I(x) is a degenerate principal ideal.
Definition 2.2 : Two ideals / 1 and / 2 of a lattice (L, :::;), are said to be disjoint , if
O is the only element common to them, i. e., / 1 n /2 = {O}.
Also, for two ideals li and 12 of a lattice (L, :S), we define li EB h ={a V b: a E
li and b E 12} which is again an ideal of L as can readily be checked.
Lemma 2.3 : Suppose I(x) and /(y) are two principal ideals of a lattice (L, :S),
where x, y E L. Then l(x) n /(y) = {O} iff x /\y = O.
Proof : Obvious.
Lemma 2.4 : Suppose I(x) and /(y) are two principal ideals of a lattice (L, :S),
where x, y E/. Then I(x) EB /(y) = L iff x Vy = 1
P roof : Obvious.
Definit ion 2.5 : A lattice (L, :S) is said to be disconnected, if there exist two nontrivial
principal ideals /(x) and /(y) of L, where x , y EL, such that I(x) n/(y) = {O}
and /(x) EB /(y) = L.
Now, for any topological space X, we take L(X ) = {A ~ X : A is closed in X}.
Then (L(X) , ~) becomes a bounded distributive lattice with X as its largest mem-
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ber and 0 as its smallest member. In the next theorem we see that the concept of
disconnectedness of X (as a topological space) is equivalent to the concept of disconnectedness
of L(X) (as a lattice).
Proposition 2.6 : A topological space X is disconnected iff the corresponding lattice
L(X) is disconnected.
Proof : Follows as a direct consequence of Lemma 2.3 and Lemma 2.4.
Proposition 2. 7 : If two spaces X and Y are homeomorphic, then the corresponding
lattices L(X) and L(Y) are isomorphic.
Proof : Suppose f : X _, Y is a homeomorphism. Define f* : L(X) _, L(Y) by
f*(A) = f (A) , for all A E L(X) . Since f is a closed map, f* is well defined. Since
f is a bijection, ¡-1 f(A) = A, for all Ar:;; X and f ¡-1(B) = B , for all B r:;; Y; and
these two facts prove that f* is bijective. That f* is order preserving is immediate,
and hence L(X) and L(Y) become isomorphic.
We now give an Example to show that the converse of Proposition 2. 7 is indeed
false.
Example 2.8 : Let X and Y be two indiscrete spaces with different cardinalities,
i.e., IXI f. IYI- Then L(X) = {X, 0} and L(Y) = {Y, 0}, and hence the map
w : L(X) _, L(Y) defined by w(X) = Y and w(0) = 0 is a lattice isomorhism. But
X and Y cannot be homeomorphic as IXI f. IYl-
In the next result we show that the converse of Proposition 2.7 holds within the
class of T1-spaces.
Proposition 2.9 : Let X and Y be two T1-spaces. If the lattices L(X) and L(Y)
are isomorphic, then the spaces X and Y are homeomorphic.
Proof : Since X and Y are T1-spaces, {x} E L(X) and {y} E L(Y), for all
x E X and y E Y. Let \]} : L(X) _, L(Y) be an isomorphism. Then w(0) =
0, w(X) = Y , and we show that \]} maps singletons into singletons. In fact, for
any x E X , w({x}) f. 0 (as w(0) = 0 and \]}is bijective ). If Y1,Y2 E w({x})
with y1 f. Y2 , then {yi} r:;; w({x}) and {y2} r:;; w({x}) which imply w-1({yi}) r:;;
w-1w({x}) = {x} and w-1({y2}) r:;; w- 1w({x}) = {x}. Since w is an isomorphism
so is w- 1 : L(Y) _, L(X), and hence {x} = w-1({yi}) f. w-1 ({y2}) = {x}, a contra-
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diction. Hence \li ( { x}) is a singleton. Thus we can define a map f : X -+ Y given
by {f(x)} = \!i({x}) for all x E X. To prove the continuity off, let FE L(Y) be
arbitrary. Sin ce \li is surjective, there exists G E L(X) such that \li ( G) = F. It is easy
to veriy that ¡-1(F) = G. Thus f becomes continuous. Since w-1 : L(Y) -+ L(X) is
also an isomorphism, the continuity of ¡-1 follows. Hence f becomes a homeomorphism.
Corollary 2.10 : Two T1 spaces X and Y are homeomorphic iff the lattices L(X)
and L(Y) are isomorphic.
proof: Follows from Proposition 2.7 and Proposition 2.9.
Remark 2.11 : The result in Proposition 2.9 appears, in essence, in Birkhoff [l].
We have preferred to give a proof in our setting.
3. LATTICE OF COMPACT SETS OF A HAUSDORFF
SPACE
In this section we shall always assume that X is a Hausdorff space. To each Hausdorff
space X we shall associate another object K(X) which consists of all compact
subsets of X , i.e, K(X) = {A ~ X : A is compact}. Then K(X) is also a lattice
under set inclusion. Since in a non-Hausdorff space, the intersection of two compact
sets may not be compact, K(X) may fail to be a lattice for a non-Hausdorff space
X. So the assumption that X is Hausdorff is necessary. Also, for a Trspace X,
K(X) ~ L(X) and consequently K(X) is a sublattice of L(X). In fact, K(X) is
an ideal in L(X) , because closed subset of a compact space is compact. The lattice
K(X) , in general, is not bounded above. Indeed, if we consider the space N of positive
integers under usual topology, then K(N) = {A~ N: A is finite} which is not
bounded above. However, it is easy to verify that K(X) is bounded above iff X is
compact.
Lemma 3.1 : Suppose X and Y are two Hausdorff spaces. Then any continuous map
f: X-+ Y induces a homomorphism K(f): K(X)-+ K(Y).
Proof: Define K(f) : K(X) -+ K(Y) by K(f)(A) = f(A), for all A E K(X). Since
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continuous image of a compact set is compact, K(f) is well-defined. Obviously, K(f)
is order preserving, and hence is a homomorphism.
We now like to establish the following functorial properties of K.
Lemma 3.2 : Suppose X, Y, Z are Hausdorff spaces and f : X ___, Y, g : Y ___, Z are
continuous maps. Then K(g0 f) = K(g) 0 K(f).
Proof: Far any A E K(X), K(g o !)(A) = (g o !)(A) = g(f(A)) = g(K(f)(A)) =
K(g)(K(f)(a)) = (K(g) o K(f))(A). Thus K(g o!) = K(g) o K(f).
Lemma 3.3 : Suppose X is a Hausdorff space and l x : X ___, X is the identity map.
Then K(lx) = lK(X) = the identity map on K(X).
Proof: For any A E K(X) , K(lx)(A) = l x (A) = A = lK(x)(A). Thus K(lx) =
lK(X)·
Proposition 3.4 : Let HTop denote the category of Hausdorff spaces and continuous
maps, and DLat denote the category of distributive lattices and homomorphisms.
Then K: HTop ___, DLat is a covariant functor.
Proof : Follows from Lemma 3.2 and Lemma 3.3.
Proposition 3.5 : If two Hausdorff spaces X and Y are homeomorphic, then K(X)
and K(Y) are isomorphic.
Proof : There is hardly anything to be proved once K is known to be a covariant
functor.
In the next example we see that the converse of the above proposition is false,
even if the underlying spaces are Hausdorff.
Example 3.6 : Let lR denote the set of reals. We define two topologies 71 and 72 on
lR as follows:
71 = {A s;:: lR : O ~ A or (JR \A) is at most countable } and 72 = the discrete topology
on R Let X = (JR, 71) and Y = (JR, 72) . Then X and Y are Hausdorff spaces. Also
K(X) = K(Y) = {As;:: lR: A is finite }. Clearly K (X) and K(Y) are isomorphic,
but X and Y cannot be homeomorphic as X is Lindelof while Y is not.
The following proposition shows that the converse of Proposition 3.5 holds within
the class of compact T2 spaces.
Proposition 3. 7 : Suppose X and Y are compact T2 spaces. Then X and Y are
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horneornorphic iff K(X) and K(Y) are isornorphic.
Proof: Suppose K(X) and K(Y) are isornorphic. Since X and Y are cornpact T2
spaces, K(X) = L(X) and K(Y) = L(Y). The rest fallows frorn Proposition 2.9, the
converse being already proved in Proposition 3.5.
4. SOME SEPARATION PROPERTIES FOR A BOUNDED
DISTRIBUTIVE LATTICE
We now introduce sorne separation properties in a bounded distributive lattice
(L, ~) and then investigate the interrelations between sorne well k.nown separation
properties like Hausdorffness, norrnality etc. of a topological space X , and the introduced
types of separation properties.
Definition 4.1 : A lattice (L , ~) (which is always assurned to be bounded and distributive)
is called
(i) a -separated, if far any two non-trivial disjoint degenerate principal ideals I(x)
and I(y) , there exisL Lwo non-trivial principal ideals I(a) and I(b) such that I(x) ~I(a),
I(y) ~I(b), x /\ b = y/\ a= O and I(a) EB I(b) = L, where x, y, a, b EL;
(ii) ,8-separated, if far any two non-trivial disjoint principal ideals I(x) and I(y) ,
there exist two non-trivial principal ideals I(a) and I(b) such that I(x) ~I(a) , I(y) ~I(b),
x /\ b = y/\ a= O and I(a) EB I(b) = L, where x, y, a, b EL;
(iii) ¡ -separated, if far any two non-trivial disjoint principal ideals I(x) and I(y) ,
there exist two non-trivial principal ideals I(a) and I(b) such that I(x) ~I(a), I(y) ~ I(b)
anda/\ b = O.
Remark 4.2 : It is clear frorn the above definition that every ,8-separated lattice is
a -separated. But the converse is false. In fact, if X is any set containing faur points,
then the bounded distributive lattice P(X) , the power set of X , under set inclusion,
is a -separated but not /3-separated. We now give an exarnple of a ¡ -separated lattice
which is not /3-separated.
Example 4.3 : Let F(N) = {A ~ N : A is finite}, where N is the set of natural
numbers. Let L = F(N) U {N}. Then (L, ~) is a bounded distributive lattice. We
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first show that (L , <;;;) is a 1-separated. For this, !et I(A), I(B) be two non-trivial
disjoint principal ideals of (L, <;;;), where A, B E L. Then 0 # A # N, 0 # B # N
and A n B = 0. Therefore, A, B E F(N) so that N \ A U B is an infinite subset of
N. We choose x ,y E N\AUB with x #y. Let C = AU {x} and D = BU {y}.
Then A~C, B~D and C n D = 0, i.e., I(A) ~ I(C), I(B) ~ D and I(C) n I(D) = {0}.
Hence (L , <;;;) becomes 1-separated. If (L, <;;;) were ,8-separated, then there would
exist E ,F E F(N) such that I(A)~I(E),I(B) ~ I(F), A n F = B n E = 0 and
I(E) EB I(F) = L. But I(E) EB I(F) = L implies that E UF = N, a contradiction,
since E and F are finite subsets of N. Thus (L , <;;;) cannot be ,8-separated.
Remark 4.4 : It is still not known to us whether there exists a bounded distribu-tive
lattice which is ,8-separated but not 1-separated. However, we shall provide a
very large class of bounded distributive lattices which are ,8-separated as well as 1-
separated. In fact, we shall show that for any connected space X , ,8-separation of
L(X) implies the 1-separation of L(X), but not conversely.
Proposition 4.5 : Suppose X is a connected topological space. If X is a Hausdorff
space, then L(X) is a-separated.
Proof : Let I(A) and I(B) be two non-trivial disjoint degenerate principal ideals of
L(X), where A,B E L(X). Then A# 0 # B and AnB = 0. We claim that A = {x} ,
for sorne x E X. If not, !et there exist x, z E A with x # z. Then 0 # {z} #A. Since
I(A) is degenerate, I(A) = {0, A}. But {z} E L(X) (as X is T2) and { z } ~ A which
implies that {z} E I(A) , a contradiction. Similarly, B = {y}, for sorne y E X. Since
A n B = 0, x #y. By Hausdorffness of X,there exist two open sets U and V in X
such that x E U, y E V and Un V = 0. Since X is connected, we have U U V ~ X.
Let C = X\ V and D = X\ U. Then C, D E L(X). Also x E U~ X \V = C
and y E V ~ X \U = D, i.e. A ~C and B ~ D. Since x tf. D and y tf. C, we have
AnD = BnC = 0. Also, A~C and B~D imply that I(A)~I(C),I(B) ~ I(D). Again,
CU D = (X\ V) u (X\ U) = X\ Un V = X\ 0 = X , and so I(C) EB I(D) = L(X).
Hence L(X) becomes an a -separated lattice.
The following example shows that the condition of connectedness in the above
· theorem cannot be suppressed.
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Example 4.6 : Let X consists of two distinct points and be endowed with the discrete
topology. Thus X is Hausdorff, but L(X) is not a -separated.
Proposition 4. 7 If X is a T1-space and L(X) is a -separated, then X is Hausdorff.
Proof: Let x ,y E X be such that x f. y. Let A = {x} and B = {y}. Then
A, B E L(X) (as X is T1) and I (A) , I (B) are two non-t rivial disjoint degenerate
principal ideals of L(X). Since L(X) is a -separated, there exist C, D E L(X ) such
that I(A) ~ I( C) , I(B) ~ I(D) , A n D = B n C = 0 and I (C) E9 I(D) = L(X); i.e,
CU D = X. Let U = X\ D and V = X\ C. Then U and V are open subsets of X.
Since A n D = B n C = 0, we have A<:;:; U and B <:;:; V , i.e, x E U and y E V, and
moreover, U n V = X \ CU D = X\ X = 0. Thus X becomes Hausdorff.
Corollary 4.8 : A connected T1-space X is Hausdorff iff L(X) is a -separated.
Proof: The proof follows from Proposition 4.5 and Proposition 4.7.
Proposition 4.9 : If a connected space X is normal, then L(X) is ,8-separated.
Proof : The proof is similar to that of Proposition4.5 and is omitted.
Proposition 4.10 : If L(X) is ,8-separated, then X is a normal space.
Proof: We omit the proof which resembles the proof of Proposition 4.7.
From the last two propositions we have:
Corollary 4.11 : A connected space X is normal iff L(X) is ,8-separated.
Proposition 4.12 : If X is a connected normal spaces, then L(X) is 1-separated.
Proof : Let I(A) and I(B) be two non-trivial disjoint principal ideals of L(X).
Then A and B are two nonempty disjoint closed subsets of X . Since X is normal,
by Urysohn's Lemma there exists a continuous map f : X --+ [O, 1] such
that f(A) = {O} and f(B) = {l}. Since X is connected, f(X) = [O, l ]. Let
C = ¡-1([0, W and D = ¡-1m, l ]). Then C, D E L(X), A ~ C, B ~ D and Cn D = 0,
i.e. I (A) ~ I( C),I(B) ~ I(D) and C n D = 0. Thus L(X) becomes 1-separated.
That the converse of the above proposition is false is shown by the following example.
In fact , we provide a connected space X for which L(X) is 1-separated but
X fails to be normal.
Example 4.13 : Let N denote the set of positive integers and T¡ denote the cofinite
topology on N. If X = (N, T¡ ) , then X is a connected, L(X) is 1-separated but X is
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not normal.
From Corollary 4.11 , Proposition 4.12 and Example 4.13 we obtain:
Corollary 4.14 : For a connected space X, L(X) is ,8-separated =? L(X) is 1-
separated, but not conversly.
5. DIMENSION OF A SPACE X AND DEPTH OF L (X)
In t his section, our only concern is to show that for any topological space X, the
dimension of X [7] can equivalently be defined in terms of a suitable lattice-theoretic
concept, termed depth, to be introduced shortly. For this we need to be endowed
with sorne appliances, clarified as follows.
We first recall that a lattice (L, ~), which is once again assumed to be bounded and
distributive, is called lower complete if any nonempty subset of L has a glb in L.
We note at t his point that for any space X , L(X) is a lower complete lattice. The
following defini t ion is also well known.
Definition 5.1 : A non-empty subset F of a lattice (L, ~) is called a filter if (i)
O ti F, (ii) a, b E F =?a n b E F and (iii) a E F and a~ b =? b E F. If the condition
(ii) is replaced by (ii)* {aa : a E J} ~ F =? a~Iªª E F, where I is an arbitrary
indexing set, then F is called a complete filter.
Whenever we talk about complete filters in the seque!, we always assume that the
underlying lattice is lower complete.
Lemma 5.2 : Suppose (L , ~) is a lower complete lattice. Then a filter F in L is
complete iff it is of the form Fa = {b E L: a~ b} for sorne a EL\ {O}.
Proof: That Fa is a complete filter for each a EL\ {O} is immediate.
Conversely, !et F be a complete filter in L. Let F = {aa: a E J}. Then a = a~Iªª E
F and so a -f. 0.It is easy to verify that F = Fa·
Definition 5.3 : A fil ter F of a lattice (L, ~) is called prime if a/\ b E F =? a E
F or b E F, where a, b E L.
Definition 5.4 : Suppose (L, ~) is a lower complete lattice.We define the depth of
a complete prime filter P in L to be the supremum of ali integers n such that there
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is a chain P0 ~ P1 ~ · · · ~ Pn = P of complete prime filters. The depth of a complete
prime fil ter P is denoted by depP. We also define the depth of the lattice L as
depL = sup{ depP : P is a complete prime fil ter in L}.
Proposition 5.5 : For any nonempty set X, the lattice (P(X), ~) has depth zero,
and if X is infinite, then P(X) always contains a lower complete sublattice L which
is of infinite depth.
Proof : Let X be any nonempty set. Then by Lemma 5.2, any complete filter of
P(X) is of the farm FA = {BE P(X) : A~ B}, far sorne A(:¡i 0) E P(X). We now
show that FA is prime iff A is a singleton. First, let FA be prime. If possible, let
x ,y E A with x #y. If A1 = A\ {x} and A2 = A\ {y}, then A1 U A2 = A E FA, but
A1 , A2 1 FA. Hence FA is not prime, a contradiction. Conversely, if A is a singleton,
then FA is obviously prime. Thus if FA and FB are two distinct complete prime filters,
no one can be contained in the other. Hence depP(X) = O.
Next, let X be an infinite set. Let S = { x1, x2 , x 3 , . .. } be a countably infinite subset
of X. For such k EN, let Ak = {x1,x2, ... ,xk} and L = {Ak: k EN}. Then
(L , ~) is a sublattice of (P(X) , ~) , where Ak ~ At +--> k '.S l, Ak U At = Amax(k,l) and
Ak n At = Amin(k,t), far all k, l EN. To show that Lis lower complete, let L' be any
nonempty subset of L. Then L' = {Ak : k E I}, far sorne nonempty subset I of N.
By well ordering priciple, I has a least element k0 , say. Then it is easy to see that
k~JAk = Ako· so Lis lower complete. For each k EN, let Fk = {Ak: k:::; l}. Then Fk
is a complete prime filter in L, far each k EN, and Fk is prime too. For, let p, q EN
be such that Ap, Aq 1 Fk. Then p, q < k. Without loss of generality, suppose p :S q.
1
Then Ap U Aq = Aq 1 Fk (as q < k), hence Fk becomes prime, far all k EN. Thus we
have an unending cha.in F1 ~ F2 ~ F3 ~ · · · of complete prime filters in L , which proves
that depL = oo.
Definition 5.6 [7] : A subset A of a topological space X is called irreducible,
if A cannot be expressed as a union of two of its proper closed subsets of X.
Definition 5. 7 [7] : The dimension of a topological space X is defined to be the
supremum of all integers n such that there is a chain Zo ~ Z1 ~ · · · ~ Zn of irreducible
closed subsets of X.
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Theorem 5.8 : Far any space X, dimX = depL(X).
Proof : We first prove the following two Lemmas.
Lemma 5.8.1 Any complete filter P of L(X) is prime iff it is of the form FA = {BE
L(X) : A~ B}, where A is an irreducible closed subset of X.
Proof : In view of Lemma 5.2, it only remains to show that FA = { B E L(X) :
A ~ B} is prime iff A is irreducible. First , let A E L(X) be such that A is irreducible.
To show that FA is prime, let BU C E FA, where B , C E L(X). Then
A~ Bu C =?A = (AnB)U(AnC) =?A = AnB or A = AnC (as A is irreducible)
=}A~ B or A~ e=} BE FA ore E FA =} FA is prime.
Conversely, let A E L(X) be such that FA is prime. We want to show that A is
irreducible. If possible, let there exist B, CE L(X) with B ~A , C~A and BU C = A.
ThenB u e E FA which implies that BE FA ore E FA (as FA is prime) , i.e. B ;;2 A
or C ;;2 A, a contradi ction. Thus A must be irreducible.
Lemma 5.8.2 : Suppose A and B are two irreducible closed subsets of X. Then
A~B iff FA~FB.
PROOF : Obvious.
Proof of the Main Theorem : From the above two Lemmas we can say that
every chain Zo ~ Z1 ~ · · · ~ Zn of irreducible closed subsets of X gives rise to a chain
Po~ P1 ~ · · · Pn ~ of complete prime filters of L(X) and conversely. Hence we conclude
that dimX = depL(X).
Corollary 5.9 : The dimension of any discrete topological space is zero.
Proof: For a discrete space X , P (X) = L(X). The rest follows from Proposition
5.5 and Theorem 5.8.
Corollary 5.10 : The depth of L(JR) ,where lR stands for the real line (with usual
topology) , is zero.
Proof: We first note that any irreducible closed subset A of lR is a singleton. In fact ,
if x , y E A with x #y (say, x < y), then choose z E lR such that x < z < y. Then we
can write A = [A\ (-oo, z)] U [A\ (z, oo)], where each of A\ (-oo, z) and A\ (z, oo)
is a non empty proper closed set. Thus A cannot be irreducible, and consequently,
each irreducible closed subset of lR is a singleton. Hence the dimension of lR is zero,so
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that in view of Theorem 5.8, the depth of L(JR) is zero.
References
[l] G.Birkhoff - Lattice Theory, Amer. Math. Soc., 1948.
[2] J. Dugundji - Topology Prentice-Hall of India, 1975.
[3] J. Harding and A. Pogel - Every lattice with 1 and O is embeddable in the lattice
oí topologies oí some set by an embedding which preserves the 1 and O, Topology
Appl., 105(2000), no. 1, 99- 101.
[4] J.L. Kelley - General Topology, D. Van Nostrand Company, Inc., 1955.
[5] A. Pultr and A. Tozzi - A note on reconstruction oí spaces and maps from lattice
data, Quaest. Math. 24(2001) , no. 1, 55- 63.
[6] H. Rasiowa - An algebraic approach to non-classical logies, North Holland, 1974.
[7] I.R. Shafarevich, - Basic Algebraic Geometry, Springer-Verlag, 1977.
Department of Pure Mathematics,
University of Calcutta,
35, Ballygunge Circular Road,
Kolkata - 700019,
India.
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