definition of extended Euclidean topological space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of Euclidean topological space.
- The reader knows a definition of product topology.
Target Context
- The reader will have a definition of extended Euclidean topological space.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(*\overline{\mathbb{R}}\): \(= \mathbb{R} \cup \{- \infty, \infty\}\) with the topology specified below
\(*\overline{\mathbb{R}}^d\): where \(d \in \mathbb{N} \setminus \{0\}\), with the product topology
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Conditions:
\(\forall S \subseteq \overline{\mathbb{R}} (S \in \{\text{ the open subsets of } \overline{\mathbb{R}}\} \iff \exists U \in \{\text{ the open subsets of } \mathbb{R}\}, \exists U' \subseteq \{- \infty, \infty\} (S = U \cup U'))\)
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2: Note
Let us see that it is indeed a topology.
\(\emptyset = \emptyset \cup \emptyset\), where the 1st \(\emptyset\) is an open subset of \(\mathbb{R}\) and the 2nd \(\emptyset\) is a subset of \(\{- \infty, \infty\}\), so, \(\emptyset\) is open on \(\overline{\mathbb{R}}\).
\(\overline{\mathbb{R}} = \mathbb{R} \cup \{- \infty, \infty\}\), where \(\mathbb{R}\) is an open subset of \(\mathbb{R}\) and \(\{- \infty, \infty\}\) is a subset of \(\{- \infty, \infty\}\), so, \(\overline{\mathbb{R}}\) is open on \(\overline{\mathbb{R}}\).
Let \(\{U_j \cup U'_j \subseteq \overline{\mathbb{R}} \vert j \in J\}\) be any set of open subsets where \(J\) is any possibly uncountable index set. \(\cup_{j \in J} (U_j \cup U'_j) = (\cup_{j \in J} U_j) \cup (\cup_{j \in J} U'_j)\), because for each \(r \in \cup_{j \in J} (U_j \cup U'_j)\), \(r \in U_j \cup U'_j\) for a \(j \in J\), so, \(r \in U_j\) or \(r \in U'_j\) for a \(j \in J\), so, \(r \in \cup_{j \in J} U_j\) or \(r \in \cup_{j \in J} U'_j\), so, \(r \in (\cup_{j \in J} U_j) \cup (\cup_{j \in J} U'_j)\); for each \(r \in (\cup_{j \in J} U_j) \cup (\cup_{j \in J} U'_j)\), \(r \in \cup_{j \in J} U_j\) or \(r \in \cup_{j \in J} U'_j\), so, \(r \in U_j\) for a \(j \in J\) or \(r \in U'_j\) for a \(j \in J\), so, \(r \in U_j \cup U'_j\) for a \(j \in J\), so, \(r \in \cup_{j \in J} (U_j \cup U'_j)\). But \(\cup_{j \in J} U_j\) is an open subset of \(\mathbb{R}\) and \(\cup_{j \in J} U'_j\) is a subset of \(\{- \infty, \infty\}\), so, \(\cup_{j \in J} (U_j \cup U'_j)\) is open on \(\overline{\mathbb{R}}\).
Let \(\{U_j \cup U'_j \subseteq \overline{\mathbb{R}} \vert j \in J\}\) be any set of open subsets where \(J\) is any finite index set. \(\cap_{j \in J} (U_j \cup U'_j) = (\cap_{j \in J} U_j) \cup (\cap_{j \in J} U'_j)\), because for each \(r \in \cap_{j \in J} (U_j \cup U'_j)\), for each \(j \in J\), \(r \in U_j \cup U'_j\), but \(r \in \mathbb{R}\) or \(r \in \{- \infty, \infty\}\), and for the former, \(r \in U_j\) for each \(j \in J\), so, \(r \in \cap_{j \in J} U_j\), and for the latter, \(r \in U'_j\) for each \(j \in J\), so, \(r \in \cap_{j \in J} U'_j\), so, anyway, \(r \in (\cap_{j \in J} U_j) \cup (\cap_{j \in J} U'_j)\); for each \(r \in (\cap_{j \in J} U_j) \cup (\cap_{j \in J} U'_j)\), \(r \in \cap_{j \in J} U_j\) or \(r \in \cap_{j \in J} U'_j\), and for the former, for each \(j \in J\), \(r \in U_j\), so, \(r \in U_j \cup U'_j\) for each \(j \in J\), so, \(r \in \cap_{j \in J} (U_j \cup U'_j)\), and for the latter, for each \(j \in J\), \(r \in U'_j\), so, \(r \in U_j \cup U'_j\) for each \(j \in J\), so, \(r \in \cap_{j \in J} (U_j \cup U'_j)\), so, anyway, \(r \in \cap_{j \in J} (U_j \cup U'_j)\). But \(\cap_{j \in J} U_j\) is an open subset of \(\mathbb{R}\) and \(\cap_{j \in J} U'_j\) is a subset of \(\{- \infty, \infty\}\), so, \(\cap_{j \in J} (U_j \cup U'_j)\) is open on \(\overline{\mathbb{R}}\).
