Find the closure , Frontier and boundary of the following set

Question

Find the closure , Frontier and boundary of the following set in $ \mathbb{R}^2$ given by

$$ S=\{(0,1) \cup (2,0) \} \cup \{(x,y): |x|+|y|< 1 \}$$

Answer:

The closure is given by

$Cl \ (A) =\{(0,1) \cup (2,0) \} \cup \{(x,y): |x|+|y| \leq 1 \}$

But I thing the closure can be

$ \{(x,y):|x|+|y| \leq 1 \} \cup \{(0,2) \}$ , because $ \ (0,1) \in \{(x,y):|x|+|y| \leq 1 \}$.

Am I right?

Also help me with the frontier and boundary of the set.

Answer 1

The frontier of $S$ is (probably, your example fits it, and it is a usage that occurs) $\operatorname{Cl}{S}\setminus S$ (all points in the closure that were not in the original set) so in this case $$\operatorname{Fr}(S) = \{(x,y): |x| + |y| =1\}\setminus \{(0,1)\}$$

clearly: $$\operatorname{Int}(S) = \{(x,y): |x| + |y| < 1\}$$

So $$\operatorname{Bd}(S) =\operatorname{Cl}(S) \setminus \operatorname{Int}(S) = \{(2,0)\} \cup \{(x,y): |x| + |y| = 1 \}$$

So all these sets can be computed/determined one we know the closure and the interior.