convergence conditions and closed maps - Soochow Journal of ...

SOOCHOW JOURNAL OF MATHEMATICS
Volume 33, No. 2, pp. 257-261, April 2007
CONVERGENCE CONDITIONS AND CLOSED MAPS
BY
ASHA GOEL AND G. L. GARG
Abstract. The main objective of this paper is to study closed mappings between
general topological spaces in terms of cluster points of inverse images of convergent
nets. We obtain a quite strong generalization to a Fuller’s result.
1. Introduction
In 1968, Fuller introduced the concept of inversely subcontinuous mappings
in terms of cluster points of inverse images of convergent nets. He then proved
the following theorem for general topological spaces X and Y .
Theorem 1.1.([2]) Let f : X → Y be a continuous and inversely subcontinuous
mapping. If the space Y is T 2 , then f is compact.
On the other hand, in 1997, we also showed the following result.
Theorem 1.2.([3]) The continuous mapping f : X → Y is closed whenever
X is countably compact, and Y is both Frechet and T 2 .
The main objective of this paper is to generalize previous theorems, and to
produce converse implications. Moreover, we introduce the concept of inversely
sequentially subcontinuous mappings in order to obtain analogue versions to Theorem
1.1 for closed mappings.
Received August 11, 2005; revised February 20, 2006; July 20, 2006.
AMS Subject Classification. 54A20, 54C10.
Key words. cluster points, nets, inversely subcontinuous, inversely cluster preserving.
257

258 ASHA GOEL AND G. L. GARG
2. Preliminaries
Along this paper, the term space means a general topological space. No
separation axioms are assumed and no mapping is assumed to be continuous or
onto, unless it is mentioned explicitly. The notation cl(A) stands for the closure
of the subset A in the space X. Besides, sequences and nets are respectively
denoted by the symbols {x n } and (x α ), and their convergence to a point x is
then denoted by {x n } → x and (x α ) → x, respectively.
A space X is said to be Frechet if any closed subsets of X can be defined by
sequences. That is, for each subset A of X, a point x lies in the cl(A) if and only
if there exists a sequence {x n } in A which converges to x. A point y is said to be
a cluster point of a sequence or net (or accumulation point in the terminology of
Dugundji [1]) if the point y lies in the closure of that sequence or net.
On the other hand, given any mapping f : X → Y between two spaces X
and Y , we say that:
(i) f is compact ([2]), if the inverse image f −1 (K) is compact for every compact
subset K ⊂ Y .
(ii) f is perfect ([3]), if it is continuous, closed and has compact fibers f −1 (y),
for each point y ∈ Y .
(iii) f is inversely subcontinuous ([2]), if any net (x α ) has a cluster point in X,
whenever its image (f(x α )) converges to a point in Y .
(iv) f is inversely cluster preserving, if for any net (x α ) whose image (f(x α ))
converges to a point y in Y , the inverse fiber f −1 (y) meets the cl((x α )).
That is, there exists at least one point x in the fiber f −1 (y) which is a
cluster point of (x α ).
We also introduce the definitions inversely sequentially subcontinuous and
inversely sequentially cluster preserving, by considering sequences {x n } instead
of nets (x α ) in previous definitions (iii) and (iv) above.
Remark 2.1. It can be easily proved that a mapping f : X → Y is closed
whenever the following condition holds: For any net (x α ) whose image (f(x α ))
converges to y in Y , there exists a point x in the inverse fiber f −1 (y) such that
(x α ) → x. Moreover, if the space Y is Frechet, we can use sequences {x n } instead
of nets (x α ) in previous statement.

CONVERGENCE CONDITIONS AND CLOSED MAPS 259
Remark 2.2. Obviously, every inversely cluster preserving mapping f is
both inversely subcontinuous and inversely sequentially cluster preserving. Besides,
inversely subcontinuous implies inversely sequentially subcontinuous. And
every mapping with compact (resp. countable compact) domain is inversely subcontinuous
(resp. inversely sequentially subcontinuous).
3. Main Results
The following two theorems are the main contribution of this paper. In
particular, Theorem 3.2 is a strong generalization to Fuller’s Theorem 1.1.
Theorem 3.1. A mapping f : X → Y between arbitrary spaces is compact
and closed if and only if it is inversely cluster preserving.
Proof. We firstly prove that f is compact. Thus, let B a compact subset
of Y , and (x α ) be any net in the inverse image f −1 (B). Notice that (f(x α )) is
also a net in the compact set B, so it must have a cluster point y ∈ B. The fact
that f is inversely cluster preserving implies there exists x ∈ X which is a cluster
point of (x α ), and such that f(x) = y. That is, the net (x α ) has a cluster point
x ∈ f −1 (B), and so f −1 (B) is a compact set and f is a compact mapping.
Now, we prove that f is closed. Let F be a closed subset of X, and y ∈
cl(f(F)). Then there exists a net (x α ) in F whose image (f(x α )) converges to
y. Since f is inversely cluster preserving, there exists x ∈ X which is a cluster
point of (x α ), and such that f(x) = y. The fact that F is closed implies that
both x ∈ F and y ∈ f(F). Hence, f(F) is a closed set and f is a closed mapping.
Conversely, suppose that f is closed and the fiber f −1 (y) is compact for every
point y ∈ Y . Besides, let (x α ) be any net in X whose image (f(x α )) converges
to a point y in Y . The fact that f is closed implies that y ∈ f(X). Now, if no
point of the inverse fiber f −1 (y) is a cluster point of the net (x α ), then for each
z ∈ f −1 (y) there exists an open neighborhood U(z) of z and an index α(z) such
that x α /∈ U(z) for every α > α(z). Notice that {U(z) : z ∈ f −1 (y)} is an open
cover of the compact set f −1 (y). Therefore, the inverse fiber f −1 (y) is contained
⋃
in the union U = n U(z j ), for a given definite set {z j }; and so there exists an
j=1
index α 0 such that x α /∈ U for every α > α 0 . Notice that y lies inside the open set

260 ASHA GOEL AND G. L. GARG
V = Y \f(X\U), and f(x α ) /∈ V for every α > α 0 , because f is a closed mapping
and f −1 (y) is contained in the open set U. Previous statement is a contradiction
to the hypothesis that (f(x α )) converges to y. Thus, there exists at least one
point x in f −1 (y) which is a cluster point of (x α ), and so f is inversely cluster
preserving.
The following theorem is a very strong generalization to Fuller’s Theorem 1.1.
We use the same hypothesis as that of Fuller, but we prove that the mapping f
is perfect, and we also get the converse implication.
Theorem 3.2. Let f : X → Y be a continuous mapping defined from the
general space X into a T 2 space Y . The mapping f is perfect if and only if f is
inversely subcontinuous.
Proof. Suppose that f is perfect, then we can prove that f is inversely
cluster preserving following the last section in the proof of Theorem 3.1; and so
f is inversely subcontinuous as well.
On the other hand, suppose that f is inversely subcontinuous. We only need
to show that f is inversely cluster preserving, in order to conclude that f is closed
and compact, according to Theorem 3.1; and so f is perfect. Let (x α ) be any net
in X whose image (f(x α )) converges to a point y in Y , then (x α ) has at least a
cluster point x in X, because f is inversely subcontinuous. Now, the fact that
f is continuous implies that f(x) is also a cluster point of (f(x α )). Since, the
space Y is T 2 and (f(x α )) converges to y, we have that y = f(x). Thus, the
continuous function f is inversely cluster preserving, and so it is closed, compact
and perfect.
Finally, recalling that closed subsets in a Frechet space can be defined via
sequences, we can prove the following result.
Theorem 3.3. Let X be an arbitrary space, and Y be Frechet. The mapping
f : X → Y is closed whenever it is inversely sequentially cluster preserving.
Proof. We show that f is closed following step by step the proof of Theorem
3.1, we only need to work with sequences instead of nets.
Finally, the following theorem is a generalization of our Theorem 1.2.

CONVERGENCE CONDITIONS AND CLOSED MAPS 261
Theorem 3.4. Let X be an arbitrary space, and Y be a Frechet T 2 space. The
mapping f : X → Y is closed whenever it is inversely sequentially subcontinuous.
Proof. In view of Theorem 3.3, we only need to prove that f is inversely
sequentially cluster preserving, in order to conclude that f is closed. We can
easily prove this fact by following step by step the proof of Theorem 3.2, we only
need to work with sequences instead of nets.
Acknowledgments
The authors are thankful to the referee for giving valuable comments.
References
[1] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966.
[2] R. V. Fuller, Relations among continuous and various non-continuous functions, Pacific J.
of Math., 25(1968), 495-509.
[3] G. L. Garg and Asha Goel, On maps: Continuous, closed, perfect, and with closed graph,
International Journal of Math. and Math. Sci., 20:2(1997), 405-408.
[4] A. Wilansky, Topology for Analysis, Xerox College Publishing, Toronto, 1970.
Department of Applied Sciences, Punjab Engineering College (Deemed University), Chandigarh
– 160 012, India.
E-mail: ashagoel30@yahoo.co.in
Department of Mathematics, Punjabi University, Patiala – 147 002, India.