Compact subsets of a topological space are used to define coc-open sets as new generalized open sets, and then coc-open sets are used to define (coc)^{∗}-open sets as another type of generalized open sets. Several results and examples related to them are obtained; particularly a decomposition of open sets is given. Also, coc-open sets and (coc)^{∗}-open sets are used to introduce coc-continuity and (coc)^{∗}-continuity, respectively. As a main result, a decomposition theorem of continuity is obtained.

Throughout this paper by a space we mean a topological space. Let

Throughout this paper, we use

A subset

The family of all

Let

By the definition one has directly that

Let

Let

The following result follows directly from Definition

Let

Let

For a space

Let

Let

Obvious.

Each of the two inclusions in Theorem

In Remark

A space

It is well known that every

Let

(a)

(b)

(c)

If _{2}-space, then

For any space

Let

For any space

Theorems

If

For every

Each of the following three examples shows that the converse of Theorem

Let

Let

Let

The following question is natural: Is there a space

The following example shows that the answer of the above question is yes.

Let

Let

Let

Let

The following result is a partial answer for Question

Let

By Theorem

Let

Let

A space

For any infinite set

Let

According to Remark

In Theorem

If

Let

In Theorem

A subset

The family of all

Let

Let

If

Let

If

According to Corollary

If

By Theorem

The following result is a new decomposition of open sets in a space.

Let

By Theorems

For a space

if

(a) The proof follows directly from Theorem

(b) Let

The following example shows that arbitrary union of

Consider the space defined in Example

A function

The following theorem follows directly from the definition.

A function

Every

Straightforward.

The identity function

The proof of the following result follows directly from Theorem

Let

The following example shows that the composition of two

Let

(a) If

(b) If

(a) It follows by noting that a function

(b) The proof follows directly from Theorem

If

Let

If

By Theorem

The following result follows directly from Theorem

Let

A function

Let

Let

Let

Let

A function

Every continuous function is

The proof follows directly from Theorem

If

The proof follows directly from Theorem

If

By Corollary

Let

The proof follows directly from Theorem

By Theorem

We end this section by the following decomposition of continuity via

A function

The proof follows directly from Theorem

The authors are very grateful to the referees for their valuable comments and suggestions. Also, the authors would like to thank Jordan University of Science and Technology for the financial assistant of this paper.