description/proof of that for finite-product metric space, topology induced by product metric is product topology of topologies induced by constituent metrics
Topics
About: metric space
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of finite-product metric space.
- The reader knows a definition of topology induced by metric.
- The reader knows a definition of product topology.
- The reader admits the proposition that for any finite-product metric space and any open ball, there is the product of some open balls contained in the open ball in which (the product) an open ball is contained.
- The reader admits local criterion for openness.
Target Context
- The reader will have a description and a proof of the proposition that for any finite-product metric space, the topology induced by the product metric is the product topology of the topologies induced by the constituent metrics.
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:
\(J\): \(\in \{\text{ the finite index sets }\}\)
\(\{M_j \vert j \in J\}\): \(M_j \in \{\text{ the metric spaces }\}\) with any metric, \(dist_j\)
\(\times_{j \in J} M_j\): \(= \text{ the finite-product metric space }\) with the finite-product metric, \(dist\)
\(O'\): \(= \text{ the topology induced by } dist\)
\(O\): \(= \text{ the product topology of the topologies induced by } dist_j \text{ s }, \{O_j \vert j \in J\}\)
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Statements:
\(O' = O\)
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2: Proof
Whole Strategy: Step 1: take any \(U' \in O'\), and see that \(U' \in O\); Step 2: take any \(U \in O\), and see that \(U \in O'\).
Step 1:
Let \(U' \in O'\) be any.
Let \(u' \in U'\) be any.
There is an open ball, \(B_{u', \epsilon} \subseteq \times_{j \in J} M_j\), such that \(B_{u', \epsilon} \subseteq U'\), by the definition of topology induced by metric.
For \(\beta := \epsilon / \sqrt{\vert J \vert}\), \(\times_{j \in J} B_{u'^j, \beta} \subseteq B_{u', \epsilon}\), by the proposition that for any finite-product metric space and any open ball, there is the product of some open balls contained in the open ball in which (the product) an open ball is contained.
But each \(B_{u'^j, \beta} \subseteq M_j\) is an open neighborhood of \(u'^j\) in the topology for \(M_j\) induced by \(dist_j\), and \(\times_{j \in J} B_{u'^j, \beta} \subseteq \times_{j \in J} M_j\) is an open neighborhood of \(u'\) in the product topology.
As \(\times_{j \in J} B_{u'^j, \beta} \subseteq B_{u', \epsilon} \subseteq U'\), \(U' \in O\), by local criterion for openness.
Step 2:
Let \(U \in O\) be any.
Let \(u \in U\) be any.
\(U = \cup_{j' \in J'} \times_{j \in J} U_{j, j'}\) where \(J'\) is a possibly uncountable index set and \(U_{j, j'} \subseteq M_j\) is open in the topology induced by \(dist_j\): refer to Note for the definition of product topology.
\(u \in \times_{j \in J} U_{j, j'}\) for a \(j' \in J'\).
\(u^j \in U_{j, j'}\), so, there is an open ball, \(B_{u^j, \epsilon_j} \subseteq M_j\), such that \(B_{u^j, \epsilon_j} \subseteq U_{j, j'}\), by the definition of topology induced by metric.
Let \(\epsilon = Min \{\epsilon_j \vert j \in J\}\).
\(B_{u^j, \epsilon} \subseteq U_{j, j'}\) and \(\times_{j \in J} B_{u^j, \epsilon} \subseteq \times_{j \in J} U_{j, j'}\).
\(B_{u, \epsilon} \subseteq \times_{j \in J} B_{u^j, \epsilon}\), by the proposition that for any finite-product metric space and any open ball, there is the product of some open balls contained in the open ball in which (the product) an open ball is contained.
So, \(B_{u, \epsilon} \subseteq \times_{j \in J} B_{u^j, \epsilon} \subseteq \times_{j \in J} U_{j, j'} \subseteq \cup_{j' \in J'} \times_{j \in J} U_{j, j'} = U\).
So, \(U \in O'\), by the definition of topology induced by metric.
