[Math] Difference between frontier and boundary points

Question

I am confused, in some books ,definitions of boundary point and frontier point is same .But some of my friends says both are different . Please explain by examples .

And also closure of a set A is union of A and it's limit points . But is there any counter example which can disprove that closure of A is not union of A and it's boundary point .

At this time according to me "a point p is boundary point of A if every neighbourhood of p contains point of A and complement of A "(may be this definition is wrong )

Answer 1

Your definition of boundary point is correct, and following that definition, the claim

For every set $A$, the closure of $A$ is the union of $A$ and the boundary of $A$

is true and therefore has no counterexample.

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As far as the term frontier goes, wikipedia explains

However, frontier sometimes refers to a different set, which is the set of boundary points which are not actually in the set; that is, $\overline S\setminus S$.

So, there are two different uses of the terms, and you just have to be careful to know which one is used in a given context. And if you are writing, when using the terms, always define them first.