definition of topology induced by pseudometric
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of pseudometric.
- The reader knows a definition of topology.
Target Context
- The reader will have a definition of topology induced by pseudometric.
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:
\( M\): \(\in \{\text{ the pseudometric spaces }\}\)
\(*O\): \(\in \{\text{ the topologies of } M\}\)
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Conditions:
\(\forall S \subseteq M (S \in O \iff (\forall p \in S (\exists \epsilon \in \mathbb{R} (0 \lt \epsilon \land (B_{p, \epsilon} \subseteq S)))))\), where \(B_{p, \epsilon}\) is the open ball around \(p\) of radius \(\epsilon\)
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2: Note
Let us see that \(O\) is indeed a topology.
\(\emptyset \in O\).
\(M \in O\).
For any \(\{U_j \vert j \in J\} \subseteq O\) where \(J\) is any possibly uncountable index set and \(U := \cup_{j \in J} U_j\), for each \(u \in U\), \(u \in U_j\) for a \(j \in J\), and there is a \(B_{u, \epsilon}\) such that \(B_{u, \epsilon} \subseteq U_j\), and \(B_{u, \epsilon} \subseteq U_j \subseteq U\), so, \(U \in O\).
For any \(\{U_j \vert j \in J\} \subseteq O\) where \(J\) is any finite index set and \(U := \cap_{j \in J} U_j\), for each \(u \in U\), \(u \in U_j\) for each \(j \in J\), and for each \(j \in J\), there is a \(B_{u, \epsilon_j}\) such that \(B_{u, \epsilon_j} \subseteq U_j\), and for \(\epsilon := min \{\epsilon_j \vert j \in J\}\), \(0 \lt \epsilon\) and \(B_{u, \epsilon} \subseteq B_{u, \epsilon_j} \subseteq U_j\) for each \(j \in J\), so, \(B_{u, \epsilon} \subseteq U\), so, \(U \in O\).
So, \(O\) is a topology.
So, \(dist\) 's being "pseudo" does not disable inducing a topology.
But of course, being "pseudo" influences some properties of the topology: for some \(m_1, m_2 \in M\) such that \(m_1 \neq m_2\) and \(dist (m_1, m_2) = 0\), any open neighborhood of \(m_1\) contains \(m_2\), so, \(M\) is not guaranteed to be (especially) Hausdorff (compare with the proposition that the topological space induced by any metric is Hausdorff).
For each \(m \in M\) and each \(\epsilon\), \(B_{m, \epsilon} \in O\), because for any \(m' \in B_{m, \epsilon}\), \(dist (m, m') \lt \epsilon\), so, \(0 \lt \epsilon - dist (m, m')\), and for \(B_{m', \epsilon - dist (m, m')}\), for each \(m'' \in B_{m', \epsilon - dist (m, m')}\), \(dist (m'', m') \lt \epsilon - dist (m, m')\), but \(dist (m'', m) \le dist (m'', m') + dist (m', m) \lt \epsilon - dist (m, m') + dist (m', m) = \epsilon\), which means that \(B_{m', \epsilon - dist (m, m')} \subseteq B_{m, \epsilon}\).
In the definition, 'open ball' can be replaced by 'open cube' without any change of the concept, because if there is an open ball, there will be an open cube contained in it, and if there is an open cube, there will be an open ball contained in it.
Sometimes, 'open cube' is more convenient, and we can safely use 'open cube' instead of 'open ball'.
