Modular Spaces Topology

In this paper, we present and discuss the topology of modular spaces using the filter base and we then characterize closed subsets as well as its regularity.


Introduction
In the theory of the modular spaces X  , the notion of ∆ 2 -condition depends on the convergence of the sequences in modular space X  .More precisely, it reads: we have . This condition has been used to study the topology of modular spaces, see J. Musielak [1], and to establish some fixed point theorems in modular spaces, see [2][3][4][5][6][7].Some fixed point theorems without ∆ 2 -condition can be found in [8,9].
In this paper, we present a new equivalent form for the ∆ 2 -condition in the modular spaces X  which is used to show that the corresponding topology is separate and to establish some associated topological properties, including the characterization of the  -closed subsets as well as its regularity.The present work is an improved English version of a pervious preprint in French [10].

Preliminaries
We begin by recalling some definitions.
Definition 2.1 Let X be an arbitrary vector space over K   or . 1) A functional     , then the modular  is called convex.
3) For given modular  in X, the is called the Luxemburg norm.

Topology τ in Modular Spaces
In this section, we introduce the property 0  for a modular  , which will be used to show that the corresponding topology, noted by , on modular space  X  is separate, and to characterize their closed subsets.
We begin with the following Proposition 3.1 Consider the family The family is a filter base. 2) Any element of  is balanced and absorbing.Furthermore, if  is convex, then any element of is convex.

1)
is a filter base.Indeed, we have and set .Then, for any we have Hence is a filter base for the existence of is balanced.Indeed, for given This means that . Whence, for all there . This shows that is absorbing.
Now, assume that  is in addition convex and let .For given Proof.In Proposition 3.1, we have seen that the family is a filter base, and furthermore any element of is balanced and absorbing.On the other hand, for any , there exists In fact, let  ; > > 0 r  .Since  satisfies the property 0  , there are and > 0 L > 0


, such that for we see that for , and so Hence the family is a fundamental system of neighborhoods of zero, then the unique topology defined by in so that X  is a topological vector space.
To show that   , X   is separate, let x, y in X  such that x y  and assume that for any V x neighborhood of x and V y neighborhood of y we have Since  satisfies the property 0  , then there exist for any 0 Thus,   0 x y    and then x = y, a contradiction since by hypothesis x y  .Therefore there exist neighbor- hoods x V of x and neighborhood y V of y such that τ Convergence and Characterization of τ-Closed

Subsets of X ρ
We begin by recalling some needed definitions of the  -convergence and the  -closed subsets of the the modular space X  (see for examples [2-8]).Definition 3.2 Let X  be a modular space.
x  in X  is said to be -con- and .x  in X  is said to be convergent to x in the sense of the topology  (or simply  -convergent) if for any > 0  there exists such that whenever .
0 Note that the property 0 > n N  is a necessary condition to show the uniqueness of the limit when exists.Thus, the  -convergence need the property 0  and it is easy to see that  -convergence and  -convergence are equivalent.
Definition 3.4 Let  be a modular satisfying the property 0 The following lemma shows that the property 0  makes sense in the theory of modular spaces.
Lemma 3.1 Let  be a modular and X  be a modular space.Then  satisfies the -condition if and only if 2   satisfies the property 0  .
Proof.To prove "if", let   n n x  be a sequence in X  such that as .This implies that for all , there exists such that for any we have and , for any .It follows to zero as n goes to  , and therefore  satisfies the 2  -condition.
For "only if", let  be a modular satisfying the 2  -condition, and suppose that there exists > 0  such that for any and for any > 0 and as .However, we have In the following theorem, we show that the  -topology and the 1  -topology are the same.Theorem 3.2 Let  be a modular satisfying the ∆ 2condition and F X   The following result is needed to show Theorem 3.2.Proposition 3.2 Let  be a modular satisfying the ∆ 2 -condition and F a  -closed subset of X  .Then is an open set of the -topology 0, 0, , > 0, such that , .
Then, for any > 0  , there exists 0 such that for every , we have Whence, making use of Proposition 3.1, we get that x F  .
Conversely, assume that F is not  -closed, then F X C  is not an open set for the  -topology.There exists then . Thence, the obtained

  
Then definitions of  -convergence and  -closed subsets of X  need the hypothesis that  satisfies the ∆ 2 -condition.
The following result shows that the modular space X  is a regular space.Theorem 3.3 Let  be a modular satisfying the ∆ 2condition, A be a  -closed subset of X  and 0 x A  .
Then there exists an open neighborhood In order to show the theorem above, we need the following result.
this implies that there exists a sequence Inversely, let x A   , then by Theorem 3.2, there ex- fore, for any 0 there exist , and . More- Suppose next that 0 and let whenever and is a closed ball of the topology  .We note by   Proof.Making appeal of Theorem 3.3, there exists y note that from Proposition 3.1, there exists a sequence Finally, we take the same arguments as in the proof of Theorem 3.