The Space of Continuous Periodic Functions Is a Set of First Category in AP ( X )

Since the last century, the study on almost periodic type functions and their applications to evolution equations has been of great interest for many mathematicians. There is a large literature on this topic. Several books are especially devoted to almost periodic type functions and their applications to differential equations and dynamical systems. For example, let us indicate the books of Amerio and Prouse [1], Bezandry andDiagana [2], Bohr [3], Corduneanu [4], Diagana [5], Fink [6], Levitan and Zhikov [7], N’Guérékata [8, 9], Pankov [10], Shen and Yi [11], Zaidman [12], and Zhang [13]. Although almost periodic functions have a very wide range of applications now, it seems that giving an example of almost periodic (not periodic) functions is more difficult than giving an example of periodic functions. Also, there is a similar problem for almost automorphic functions. In this paper, we aim to compare the “amount” of almost periodic functions (not periodic) with the “amount” of continuous periodic functions, and we also discuss the related problems for almost automorphic functions.


Introduction
Since the last century, the study on almost periodic type functions and their applications to evolution equations has been of great interest for many mathematicians.There is a large literature on this topic.Several books are especially devoted to almost periodic type functions and their applications to differential equations and dynamical systems.For example, let us indicate the books of Amerio and Prouse [1], Bezandry and Diagana [2], Bohr [3], Corduneanu [4], Diagana [5], Fink [6], Levitan and Zhikov [7], N'Guérékata [8,9], Pankov [10], Shen and Yi [11], Zaidman [12], and Zhang [13].
Although almost periodic functions have a very wide range of applications now, it seems that giving an example of almost periodic (not periodic) functions is more difficult than giving an example of periodic functions.Also, there is a similar problem for almost automorphic functions.In this paper, we aim to compare the "amount" of almost periodic functions (not periodic) with the "amount" of continuous periodic functions, and we also discuss the related problems for almost automorphic functions.

Main Results
Throughout the rest of this paper, we denote by R the set of real numbers, by  a Banach space, and by (R, ) the set of all continuous functions  : R → .Definition 1 (see [4]).A function  ∈ (R, ) is called almost periodic if, for every  > 0, there exists () > 0 such that every interval of length () contains a number  with the property that sup ∈R      ( + ) −  ()     < .
We denote the collection of all such functions by ().
Recall that () is a Banach space under the supremum norm.
Here,  is called a period of .We denote the collection of all such functions by ().For  ∈ (), we call  0 the fundamental period if  0 is the smallest period of .
Remark 3. Similar to the proof in [4, page 1], it is not difficult to show that if  ∈ () is not constant, and then  has the fundamental period.
Definition 4 (see [8]).A function  ∈ (R, ) is called almost automorphic if, for every real sequence (  ), there exists a subsequence (  ) such that is well defined for each  ∈ R and lim for each  ∈ R. Denote by () the set of all such functions.
Recall that there exists an almost automorphic function which is not almost periodic, for instance, the following function: Before the proof of our main results, we need to recall the notion about the first category.
Definition 5 (see [14]).Let  be a topological space.A set  ⊂  is said to be nowhere dense if its closure has an empty interior.The sets of the first category in  are those that are countable unions of nowhere dense sets.Any subset of S that is not of the first category is said to be of the second category in .

Theorem 6. 𝑃(𝑋) is a set of first category in 𝐴𝑃(𝑋).
Proof.For  = 1, 2, . .., we denote Then, it is easy to see that We divide the remaining proof into two steps. Step where ‖ − ‖ (R) <  was used.So, we know that (, ) ⊂ () \   , which means that   is a closed subset of ().
Step 2. Every   has an empty interior.
Case I.  is constant.
Case II. is not constant.By Remark 3,  has a fundamental period  0 .We denote where   = sup ∈R ‖()‖.Obviously,   ∈ (, ).Also, we claim that   ∉   .In fact, if this is not true, then there exists  ∈ [,  + 1] such that that is, Let Then  1 () ≡  2 ().If  1 () ≡  2 () ≡ , where  is a fixed constant, then which yields since  is bounded.Thus, we have Noting that  0 is the fundamental period of  and  0 is the fundamental period of (⋅/), there exist two positive integers  and  such that that is,  = /, which is a contradiction.If  1 =  2 is not constant, then, by Remark 3, we can assume that  0 is the fundamental period of  1 and  2 .Noting that  0 is a period of  1 and  0 is a period of  2 , similar to the above proof, we can also show that  is a rational number, which is a contradiction.
In conclusion, () is countable unions of closed subsets with empty interior.So () is a set of first category.Remark 7. Since () is a set of second category, it follows from Theorem 6 that () \ () is a set of second category, which means that, to some extent, the "amount" of almost periodic functions (not periodic) is far more than the "amount" continuous periodic functions.
Remark 9.By Theorem 8, () \ () is a set of second category in (), which means that, to some extent, the "amount" of almost automorphic functions (not almost periodic) is far more than the "amount" of almost periodic functions.