The neutrality’s origin, character, and extent are studied in the Neutrosophic set. The neutrosophic set is an essential issue to research since it opens the door to a wide range of scientific and technological applications. The neutrosophic set can find its spot to research because the universe is filled with indeterminacy. Neutrosophic set is currently being developed to express uncertain, imprecise, partial, and inconsistent data. Truth membership function, indeterminacy membership function, and falsity membership function are used to express a neutrosophic set in order to address uncertainty. The neutrosophic set produces more rational conclusions in a variety of practical problems. The neutrosophic set displays inconsistencies in data and can solve real-world problems. We are directed to do our work in semi-continuous and almost continuous mapping on the basis of the neutrosophic set by observing these. Since we are going to study the properties of semi continuous and almost continuous mapping, we present the meaning of

After Zadeh [

In this current decade, neutrosophic environments are mainly interested by different fields of researchers. In Mathematics also much theoretical research has been observed in the sense of neutrosophic environment. It will be necessary to carry out more theoretical research to establish a general framework for decision-making and to define patterns for complex network conceiving and practical application. Salama et al. [

A neutrosophic set (NS) ^{-}0, 1^{+}[. Note that

Complement of

Let

(i)

(ii)

(iii)

Let

(i)

(ii)

(iii)

Then the pair

Let

Let

Let

Let

Let

Prove is Straightforward.

Let

Let

Let

For each

Let

For each

For a family

For a NS

Prove is Straightforward.

The statements below are equivalent:

(i) and (ii) are equivalent follows from Lemma 3.8, since for a NS

(i)

(iii)

(ii)

Arbitrary union of NSOSs is a NSOS, and

Arbitrary intersection of NSCoSs is a NSCoS.

(i) Let

(ii) Let

It is clear that every neutrosophic open set (NOS) (neutrosophic closed set (NCoS)) is a NSOS (NSCoS). The converse is false, it is seen in

Further, the closure of NOS is a NSOS and the interior of NCoS is a NSCoS.

Let

Then,

Let

If

Let

It is sufficient to prove

Then

We have,

A NS

A NS

A NS

It is obvious that every NROS (NRCoS) is NOS (NCoS). The converse need not be true. For this we cite an example.

From

The union (intersection) of any two NROSs (NRCoS) need not be a NROS (NRCoS).

Let

Then

Clearly,

Similarly,

Now,

But

Hence,

(i) The intersection of any two NROSs is a NROS, and

(ii) The union of any two NRCoSs is a NRCoS.

(i) Let

(ii) Let

(i) The closure of a NOS is NRCoS, and

(ii) The interior of a NCoS is NROS.

(i) Let

(ii) Let

Let

Let

Let

Let

Let

Let

From

Let

Then

Let

Then

Let

Let

That

Let

For a NOS

Let

From

The converse of

A mapping

Let

Consider that

(a)

(c)

(b)

Clearly, a NCM is NACM. But the converse needs not be true.

Let

Then

Now, let

Here,

A NTS

Let

From

which shows that

Let

Let

Now,

Thus, by

Let

Since

Let

Consider that

Thus, by

Conversely, let

Since

and hence using

Thus, by

The truth membership function, indeterminacy membership function, and falsity membership function are all employed in the Neutrosophic Set to overcome uncertainty. First, we developed the definitions of