Is it true that $A=int(A)$ or $A=int(A) \cup fr(A)$ and true that $A^C=ext(A)$ or $A^C=ext(A) \cup fr(A)$?

Question

I'm trying to understang some topology of $\mathbb{R}^n$ space.

I want to know if it's true that $A=int(A)$ or $A = \text{int}(A)\cup \partial A$ and true that $A^{c} = \text{ext}(A)$ or $A^{c} = \text{ext}(A) \cup\partial A$

Further more, if $A = \text{int}(A)\cup\partial A$, then $A^{c} = \text{ext}(A)$ and if $A = \text{int}(A)$, then $A^{c} = \text{ext}(A)\cup \partial A$

Answer 1

Given a subset $A\subseteq\mathbb{R}^{n}$, the collection $\{\text{int}(A),\partial A,\text{ext}(A)\}$ is a partition of $\mathbb{R}^{n}$.

We say that $A$ is open iff $A\cap\partial A = \varnothing$, which is equivalent to say that $A = \text{int}(A)$.

We say that $A$ is closed iff $A\cap\partial A = \partial A$, which is equivalent to say that $A = \text{int}(A)\cup\partial A$.

Finally, the relation $A^{c} = \text{ext}(A)$ holds iff $A$ is closed.

This is because $\overline{A} = \text{int}(A)\cup\partial A$.

If you still have any questions, please let me know.

Answer 2

In general, $A = int (A)$ does not hold. When it holds, we say that $A$ is an open set.

In general, $A=int(A) \cup fr(A)$ does not hold. When it holds, we say that $A$ is a closed set.

It is not always true that $A^C = ext(A)$ or that $A^C=ext(A) \cup fr(A)$.

I'm not sure that you actually posted a question.