EDB — 0NX

[0NX] For the following exercises we define that

1. a set \(E\) is open if

   \begin{equation} \label{eq:defaperti} ∀ x_ 0 ∈ E , ∃ r>0: B(x_ 0,r)⊆ E\quad . \end{equation}

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   It is easily seen that \(∅,X\) are open sets; that the intersection of a finite number of open sets is an open set; that the union of an arbitrary number of open set is an open set. So these open sets form a topology.

2. The interior \({{E}^\circ }\) of a set \(E\) is

   \begin{equation} \label{eq:internooperativo} {{E}^\circ }=\bigl\{ x∈ E: \exists r>0, B_ r(x)⊆ E \bigr\} \ ; \end{equation}

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   It is easy to verify that \({{E}^\circ }⊆ E\), and that \(E\) is open if and only if \({{E}^\circ }=E\) (exercise [0PB]↺↻).

3. A set is closed if the complement is open.

4. A point \(x_ 0∈ X\) is adherent to \(E\) if

   \[ ∀\, r{\gt}0\ ,\quad E∩ B_ r(x_ 0)≠∅ \quad . \]

5. The closure \(\overline E\) of \(E\) is the set of adherent points; it is easy to verify that \({E}⊆ \overline{E}\); It is shown that \(\overline E=E\) if and only if \(E\) is closed (exercise [0PM]↺↻).

6. \(A\) is dense in \(B\) if \(\overline A ⊇ B\), that is, if for every \(x ∈ B\) and for every \(r {\gt} 0\) the intersection \(B_ r (x) ∩ A\) is not empty.