1. Introduction
Let
be a covariant functor acting on a class of topological spaces. The following natural general problem in the theory of covariant functors was posed by V. V. Fedorchuk at the Prague Topological Symposium in 1981 (see [
1]):
Let be a topological property and a covariant functor. If a topological space X has the property , then whether has the same property, and vice versa, if has the property , does the space X also have the property ?
This paper deals with such questions.
Let
G be a subgroup of the symmetric group
,
, of all permutations of the set
, and let
X be a topological space. On the space
, define the following equivalence relation
: for elements
and
in
The relation is called the G-symmetric equivalence relation. The equivalence class of an element is denoted by or . The quotient space (equipped with the quotient topology of the topology on ) is called the space of G-permutation degree of X and is denoted by . The quotient mapping of to this space is denoted by ; when , one writes .
Let
be a continuous mapping. Define the mapping
by
It is easy to verify that as defined is a functor in the category of compacta. This functor is called the functor of G-permutation degree.
In [
1,
2], V. V. Fedorchuk and V. V. Filippov investigated the functor of
G-permutation degree, and it was proved that this functor is a normal functor in the category of compact spaces and their continuous mappings.
In recent years, a number of studies have investigated various covariant functors, in particular the functor of
G-permutation degree, and their influence on some topological properties (see, for instance, [
3,
4,
5,
6]). In [
3,
4], the index of boundedness, uniform connectedness, and homotopy properties of the space of
G-permutation degree have been studied, and it was shown in [
4] that the functor
preserves the homotopy and the retraction of topological spaces. References [
5,
6] deal with certain tightness-type properties and Lindelöf-type properties of the space of
G-permutation degree.
The current paper is devoted to the investigation of some classes of topological spaces (such as developable spaces, Moore spaces, -spaces, -spaces, Lašnev’s and Nagata’s spaces) in the space of G-permutation degree.
Throughout the paper, all spaces are assumed to be .
Observe that the space
is related to the space
of nonempty
-element subsets of
X equipped with the Vietoris topology whose base form the sets of the form
where
are open subsets of
X [
2].
Observe that the mapping
assigning to each
G-symmetric equivalence class
the hypersymmetric equivalence class
containing it represents the functor
as the factor functor of the functor
[
1,
2].
Also, the spaces
and
are homeomorphic, while it is not the case for
[
2].
2. Results
In this section, we present the results obtained in this study.
For an open cover of a space X and a subset A of X, the star of Awith respect to is defined by .
Let be an open cover of X. Obviously, is an open cover of .
Proposition 1. Let be an open cover of . For each , we have Proof. Let . Then, there exists such that . On the other hand, if and only if and for every , , there exists a permutation such that . Hence, we obtain that . This means that . □
Lemma 1. Let be points of X. For each , let be a decreasing sequence of nonempty subsets of X such that . Then, Proof. Let , and assume that . Then, for each positive integer m, . This means that there exists a permutation such that for all . In addition, for all . Consequently, it follows that . This means that . □
Proposition 2. Let X be a space, and let be points of X. For each , let be a local base of X at . Then, is a local base of at .
Proof. Without loss of the generality, suppose that for every positive integer m. Let be an open subset of which contains . Then, there exist open subsets of X such that . Put for every . Then, are open subsets of X such that . Since is a local base at , there exists a positive integer such that . Let . Then, . Consequently, and . Therefore, is a local base of at . □
A space
X is
developable [
7,
8] if there exists a sequence
of open covers of
X such that, for each
,
is a local base at
x. Such a sequence of covers is called a
development for
X. It is well known that every metrizable space is developable, and every developable space is clearly first countable.
Remark 1. Clearly, the above definition of the developable space is equivalent to the following:
(a) For each and for each positive integer m such that , is a neighborhood of the point x, and
(b) For each and for each open U containing x, there exists a positive integer m such that .
Theorem 1. If X is a developable space, then so is .
Proof. Assume that
X is a developable space and
is a development for
X. For every
, let
Then,
is also a development for
X such that
for all
and every
. Put
It can be easily checked that is an open cover of for every .
Now, we will prove that for each
,
is a local base at
. Let
be an open subset of
such that
. Then, there exist open subsets
of
X such that
. Since
is a local base at
for any
, there exists a positive integer
such that
. Then, there exists
such that
for all
. By Proposition 1, we have
By Statement (b) of Remark 1, it means that is a developable space. □
A regular developable space is a
Moore space [
7,
8].
Proposition 3. If X is a Moore space, then so is .
Proof. By Theorem 1, if
X is a developable space, then the space
is also developable. On the other hand, it is well known from [
9] that regularity is preserved under the closed-and-open mapping and Cartesian product. Therefore, if
X is a regular space, then the space
is also regular. □
A family
of subsets of a topological space is
closure preserving [
7,
9] if
for every
.
Theorem 2. If is a closure-preserving family of subsets of X, then is a closure-preserving family of subsets of .
Proof. Let be a subfamily of and . Let . Since is a closure preserving family of subsets of X, we have that . This means that is an open subset of X and for all . Let . Then, is open subset of , and for all . Therefore, . It shows that . Hence, is a closure preserving family of subsets of . □
A family
is called
σ-closure preserving [
7] if it is represented as a union of countably many closure preserving subfamilies.
An
-space [
7,
8] is a regular space having a
-closure preserving base.
Example 1. Let denote the set of rational numbers. For , put and . Define a base for a topology on X as follows: for any and such that , we put , and let be the set of all such . For all and such that and , we put, and let be the set of all . Now, put . Then one can check that is a σ-closure preserving base for X. It shows that X is an -space. Moreover, the space X is a first countable, but non-metrizable space. Theorem 3. If X is an -space, then so is .
Proof. Let X be an -space and be a -closure preserving base in X. Since the union of two closure preserving family of subsets of X is also closure preserving, we assume that for each i. For every positive integer i, set . Obviously, for all positive integers i. By Theorem 2, is a closure preserving family of subsets of , and at the same time is a family of open subsets of . Therefore, is a -closure preserving family of open subsets of .
Now, we will show that is a base for . Let be an arbitrary element of and be an open subset of such that . Since is a base for X, there exist such that . Since for each positive integer i, there exists such that . Then it follows that . Therefore, is a base for . This means that is an -space. □
A collection
of (not necessarily open) subsets of a regular space
X is a
quasi-base in
X [
7] if whenever
and
U is a neighborhood of
x, there exists a
such that
.
An
-space [
7,
8] is a regular space having a
-closure preserving quasi-base.
Theorem 4. If X is an -space, then so is .
Proof. Suppose that X is an -space and is a -closure preserving quasi-base. Since the union of two closure-preserving family of subsets of X is also closure preserving, we assume that for each i. For each positive integer i, put . Obviously, for all i. By Theorem 2, is a closure preserving family of subsets of . Therefore, is a -closure preserving family of subsets of .
Now, we will prove that is a quasi-base for . Let be an arbitrary element of and be an open subset of such that . Consequently, there exist open subsets of X such that . Since is a quasi-base for X, there exist a permutation and such that , where . Note that . It shows that is a quasi-base for . □
Recall now that a space
X is said to be stratifiable if f for every closed subset
there is a sequence of open subsets
such that (i)
, and (ii) if
, then
for each
. In the paper [
10] it was proved that a space is stratifiable if and only if it is
. Therefore, we obtain the following:
Corollary 1. If a space X is stratifiable, then so is .
A space
X is a
Lašnev space [
7,
8] if there exist a metric space
Z and a continuous closed mapping from
Z onto
X. Lašnev spaces are known to be
-spaces.
Theorem 5. Let X be a space, and let n be a positive integer. If is a Lašnev space, then so is .
Proof. Suppose that is a Lašnev space. Then, there exist a metric space Z and a continuous closed mapping . Since is a closed, onto mapping, we obtain that the mapping is also a closed mapping from the metric space Z onto the space . This means that the space is a Lašnev space. □
Theorem 6 ([
8])
. Let X be a space. Then, is a Lašnev space if and only if is a Lašnev space. As we said in the Introduction, in Reference [
2], it was shown that the spaces
and
are homeomorphic. Hence, we obtain the following corollary.
Corollary 2. Let X be a space. Then, is a Lašnev space if and only if is a Lašnev space.
A space
X is a
Nagata space [
11] provided that for each
, there exist sequences
and
of open neighborhoods of
x such that for all
:
- (1)
is a local base at x;
- (2)
if , then (or equivalently, if , then ).
The definition of the Nagata space is equivalent to the following [
11,
12]: a Nagata space is a first countable stratifiable space.
Corollary 3. Let X be a space, and let n be a positive integer. If X is a Nagata space, then so is .