EDB — 0H7

[0H7] Topics:boundary. Let \(A⊂ X\). Let’s remember the definition of boundary \(∂ A=\overline A⧵ {{A}^\circ }\). Note that \(∂ A\) is closed: indeed setting \(B=A^ c\) to be the complement, it is easily verified that \(∂ A=\overline A∩ \overline B\). In particular we showed that \(∂ A=∂ B\).

Show that the three sets \(∂ A,{{A}^\circ },{{B}^\circ }\) are disjoint, and that their union is \(X\); in particular, show that the three sets are characterized by these three properties:

• Each neighborhood of \(x\) intersects both \(A\) and \(B\);

• there exists a neighborhood \(x\) contained in \(A\);

• there exists a neighborhood \(x\) contained in \(B\).

(See also [0Q3]↺↻ for the case of metric spaces).