Chapter 3 Chapter 03 name

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3.1 Post After verify

  1. Interior of a set is open in X: The interior of a set \(S\), denoted by \(\text{int}(S)\), is defined as the union of all open sets contained in \(S\). Since the union of open sets is open, \(\text{int}(S)\) is open in \(X\).

  2. If \(T\) is open in \(X\) then \(T \subseteq S\) if and only if \(T \subseteq \text{int}(S)\): If \(T\) is an open set and \(T \subseteq S\), then every point in \(T\) is an interior point of \(S\). Therefore, \(T \subseteq \text{int}(S)\). Conversely, if \(T \subseteq \text{int}(S)\), then \(T\) is a subset of \(S\).

  3. Interior of \(S\) is an open subset of \(S\) when \(S\) is given the subspace topology: The interior of a set \(S\), \(\text{int}(S)\), is the union of all open sets contained in \(S\). Therefore, \(\text{int}(S)\) is an open subset of \(S\).

  4. \(S\) is an open subset of \(X\) if and only if \(\text{int}(S) = S\): If \(S\) is open, then every point of \(S\) is an interior point of \(S\), so \(\text{int}(S) = S\). Conversely, if \(\text{int}(S) = S\), then \(S\) is open because the interior of a set is always open.

  5. Interior operator preserves/distributes over binary intersection: The interior of the intersection of two sets \(S\) and \(T\), \(\text{int}(S \cap T)\), is equal to the intersection of the interiors of \(S\) and \(T\), \(\text{int}(S) \cap \text{int}(T)\). This is because the intersection of two open sets is open, and the interior of a set is the union of all open sets contained in it.

  6. The interior operator does not distribute over unions: The interior of the union of two sets \(S\) and \(T\), \(\text{int}(S \cup T)\), is a superset of the union of the interiors of \(S\) and \(T\), \(\text{int}(S) \cup \text{int}(T)\). This is because the union of two open sets is open, and the interior of a set is the union of all open sets contained in it. However, the equality might not hold in general. For example, if \(X = \mathbb{R}\), \(S = (-\infty, 0]\), and \(T = (0, \infty)\), then \(\text{int}(S) \cup \text{int}(T) = (-\infty, 0) \cup (0, \infty) = \mathbb{R} \setminus \{0\}\), which is a proper subset of \(\text{int}(S \cup T) = \text{int}(\mathbb{R}) = \mathbb{R}\).