Primal Spaces and Quasihomeomorphisms
Primal Spaces and Quasihomeomorphisms
Primal Spaces and Quasihomeomorphisms
2(2015), 109-118
doi:10.4995/agt.2015.3045
c AGT, UPV, 2015
Abstract
In [3], the author has introduced the notion of primal spaces. The
present paper is devoted to shedding some light on relations between
quasihomeomorphisms and primal spaces.
Given a quasihomeomorphism q : X → Y , where X and Y are prin-
cipal spaces, we are concerned specifically with a main problem: what
additional conditions have to be imposed on q in order to render X
(resp. Y ) primal when Y (resp. X) is primal.
1. Introduction
First, we recall some notions which were introduced by the Grothendieck
school (see for example [6] and [7]) such as locally closed sets and quasihomeo-
morphisms.
Let X be a topological space and S be a set of X. S is called locally closed
if it is an intersection of an open set and a closed set of X. We denote by L(X)
the set of all locally closed sets of X.
Given topological spaces X and Y , a continuous map q : X → Y is called
a quasihomeomorphism if A 7→ q −1 (A) defines a bijection from O(Y ) (resp.,
F (Y ), resp., L(Y )) to O(X) (resp., F (X), resp., L(X)) where O(X) (resp.,
F (X), resp., L(X)) is the family of all open (resp., closed, resp., locally closed)
sets of X.
On the other hand, another definition of quasihomeomorphism is given by
K.W.Yip in [9] as follows. A continuous map q : X → Y between topological
f
X −→ X
q↓ ↓q
g
Y −→ Y
That is g ◦ q = q ◦ f . For more details, one may see [4] and [5].
Let (X, f ) be a flow in the category of sets noted Set. O.Echi has defined
the topology P(f ) on X with closed sets exactly those A which are f -invariant
(A set A of X is called f -invariant if f (A) ⊆ A). Clearly P(f ) provides a
principal topology on X.
We can easily see that for any set A of X, the closure A is exactly ∪[f n (A) :
n ∈ N] and in particular for any point x ∈ X, {x} = {f n (x), n ∈ N} denoted
Of (x) and called the orbit of x by f .
The family x ↑ = {y ∈ X : f n (y) = x for some n ∈ N} is a basis of open sets
of P(f ).
According to O.Echi, a primal space is a topological space (X, τ ) such that
there is some mapping f : X → X with τ = P(f ) (for more informations see
[3]).
In the first section of this paper, we are interested in some dynamical prop-
erties of quasihomeomorphisms between principal spaces.
The main goal of the second section is to show that given an onto quasihome-
omorphism from a primal space X to a principal space Y , then Y is primal
(see Theorem 3.2).
In the third section, we move our focus to one-to-one quasihomeomorphisms
and its effects on primal spaces (see Theorem 4.1).
Finally, some particular cases of quasihomeomorphisms are studied and com-
mented.
2. Preliminary results
Let X be a principal space and ≤ its specialization quasi-order.
A point x ∈ X is called minimal if it satisfies the property:
for each y ∈ X, y ≤ x ⇒ x ≤ y.
Let f : X → Y be a continuous map between two principal spaces. It
follows immediately, from the fact that a map between two principal spaces is
continuous if and only if the induced map between the associated quasi-ordered
sets is isotone, that for every x, y ∈ X, we have:
x ≤ y ⇒ f (x) ≤ f (y).
Now, the following proposition shows that the converse holds if f is a quasi-
homeomorphism.
Proposition 2.1. Let f : X −→ Y be a quasihomeomorphism where X and Y
are principal spaces. Then, for every x, y ∈ X, we have:
x ≤ y ⇐⇒ f (x) ≤ f (y).
Proof. It is sufficient to show the second implication.
For that, let x, y ∈ X such that f (x) ≤ f (y). Since f is a quasihomeomorphism,
then there exists a unique closed set F of Y such that ↓ y = f −1 (F ).
Now, the fact that f (y) ∈ F implies ↓ f (y) ⊆ F and consequently f (x) ∈ F ,
that is x ∈↓ y as desired.
This result leads to the following corollary.
Corollary 2.2. Let q : X −→ Y be a quasihomeomorphism where X and Y are
principal spaces. For every x ∈ X, if q(x) is minimal in Y then x is minimal
in X.
Proof. Let x be a point in X such that q(x) is a minimal point in Y . Suppose
that there exists x′ ∈ X satisfying x′ ≤ x. Then, we have q(x′ ) ≤ q(x) and
thus, by minimality of q(x), q(x) ≤ q(x′ ). Therefore, Proposition 2.1 does the
job.
Question 2.3. Let q : X −→ Y be a quasihomeomorphism and x a point in
X. If x is minimal, then what about q(x) ?
The following proposition shows that if in addition q is an onto quasihome-
omorphism from X to Y , then there is an equivalence between x is minimal in
X and q(x) is minimal in Y .
We state a useful remark.
Remark 3.6. The condition ”Y is a T0 −space” in Proposition 3.5 is a sufficient
condition but not a necessary condition. To see this, consider the example
3.3.(4).
References
[1] K. Belaid, O. Echi and S. Lazaar, T(α,β) -spaces and the Wallman Compactification,
Int. J. Math. Math. Sc. 68 (2004), 3717–3735.
[2] O. Echi, Quasi-homeomorphisms, Goldspectral spaces and Jacspectral spaces, Boll.
Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)6 (2003), 489–507.
[3] O. Echi, The category of flows of Set and Top, Topology Appl. 159 (2012), 2357–2366.
[4] J. F. Kennisson, The cyclic spectrum of a Boolean flow, Theory Appl. Categ. 10 (2002),
392–409.
[5] J. F. Kennisson, Spectra of finitly generated Boolean flow, Theory Appl. Categ. 16
(2006), 434–459.
[6] A. Grothendieck and J. Dieudonné, Eléments de Géométrie Algébrique, Die Grundlehren
der mathematischen Wissenschaften, vol. 166, Springer-Verlag, New York, 1971.
[7] A. Grothendieck and J. Dieudonné, Eléments de Géométrie Algébrique. I. Le langage
des schḿas, Inst. Hautes Études Sci. Publ. Math. No. 4, 1960.
[8] M. H. Stone, Applications of Boolean algebra to topology, Mat. Sb. 1 (1936), 765–772.
[9] K. W. Yip, Quasi-homeomorphisms and lattice-equivalences of topological spaces, J.
Austral. Math. Soc. 14 (1972), 41–44.
[10] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, Con-
tinuous Lattices and Domains, Cambridge Univ, Press, 2003.