description/proof of that for metric space with induced topology and positive real number, distance as minimum of original distance and number is metric and induces original topology
Topics
About: topological space
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of topology induced by metric.
Target Context
- The reader will have a description and a proof of the proposition that for any metric space with the induced topology and any positive real number, the distance as the minimum of the original distance and the number is a metric and induces the original topology.
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:
\(M\): \(\in \{\text{ the metric spaces }\}\) with metric, \(dist'\), with the induced topology, \(O'\)
\(r\): \(\in \mathbb{R}\) such that \(0 \lt r\)
\(dist\): \(: M \times M \to \mathbb{R}, (m_1, m_2) \mapsto Min (\{dist' (m_1, m_2), r\})\)
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Statements:
\(dist \in \{\text{ the metrics for } M\}\)
\(\land\)
\(\text{ the topology induced by } dist, O = O'\)
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2: Proof
Whole Strategy: Step 1: see that \(dist\) is a metric; Step 2: see that for each \(\epsilon \lt r\) and each \(m \in M\), \(B_{m, \epsilon} = B'_{m, \epsilon}\) where \(B_{m, \epsilon}\) and \(B'_{m, \epsilon}\) are the open ball by \(dist\) and \(dist'\), respectively; Step 3: see that for each \(U' \in O'\), \(U' \in O\); Step 4: see that for each \(U \in O\), \(U \in O'\); Step 5: conclude the proposition.
Step 1:
Let us see that \(dist\) is a metric.
For each \(m_1, m_2, m_3 \in M\), 1) \(0 \le dist (m_1, m_2)\) and \(dist (m_1, m_2) = 0 \iff m_1 = m_2\): when \(dist (m_1, m_2) = 0\), \(dist (m_1, m_2) = dist' (m_1, m_2) = 0\), so, \(m_1 = m_2\); when \(m_1 = m_2\), \(dist (m_1, m_2) = dist' (m_1, m_2) = 0\); 2) \(dist (m_1, m_2) = dist (m_2, m_1)\): \(dist (m_2, m_1) = Min (\{dist' (m_2, m_1)\}, r) = Min (\{dist' (m_1, m_2)\}, r) = dist (m_1, m_2)\); 3) \(dist (m_1, m_3) \le dist (m_1, m_2) + dist (m_2, m_3)\): when \(dist (m_1, m_3) = dist' (m_1, m_3)\), \(dist (m_1, m_3) = dist' (m_1, m_3) \le dist' (m_1, m_2) + dist' (m_2, m_3)\), and when \(dist (m_1, m_2) = dist' (m_1, m_2)\) and \(dist (m_2, m_3) = dist' (m_2, m_3)\), \(dist (m_1, m_3) = dist' (m_1, m_3) \le dist' (m_1, m_2) + dist' (m_2, m_3) = dist (m_1, m_2) + dist (m_2, m_3)\) and otherwise, \(dist (m_1, m_3) \le r \le dist (m_1, m_2) + dist (m_2, m_3)\); when \(dist (m_1, m_3) = r\), \(dist (m_1, m_3) = r \le dist' (m_1, m_3) \le dist' (m_1, m_2) + dist' (m_2, m_3)\), and when \(dist (m_1, m_2) = dist' (m_1, m_2)\) and \(dist (m_2, m_3) = dist' (m_2, m_3)\), \(dist (m_1, m_3) \le dist' (m_1, m_2) + dist' (m_2, m_3) = dist (m_1, m_2) + dist (m_2, m_3)\) and otherwise, \(dist (r_1, r_3) = r \le dist (m_1, m_2) + dist (m_2, m_3)\).
Step 2:
Let \(\epsilon \in \mathbb{R}\) be any such that \(\epsilon \lt r\).
Let \(m \in M\) be any.
Let \(B_{m, \epsilon}\) be the open ball by \(dist\).
Let \(B'_{m, \epsilon}\) be the open ball by \(dist'\).
Let us see that \(B_{m, \epsilon} = B'_{m, \epsilon}\).
Let \(b \in B_{m, \epsilon}\) be any.
As \(dist (m, b) = Min (\{dist' (m, b), r\}) \lt \epsilon \lt r\), \(dist' (m, b) \lt \epsilon\), so, \(b \in B'_{m, \epsilon}\).
So, \(B_{m, \epsilon} \subseteq B'_{m, \epsilon}\).
Let \(b' \in B'_{m, \epsilon}\) be any.
As \(dist' (m, b') \lt \epsilon \lt r\), \(dist (m, b') = Min (\{dist' (m, b'), r\}) = dist' (m, b') \lt \epsilon\), so, \(b' \in B_{m, \epsilon}\).
So, \(B'_{m, \epsilon} \subseteq B_{m, \epsilon}\).
So, \(B_{m, \epsilon} = B'_{m, \epsilon}\).
Step 3:
Let \(U \in O\) be any.
Let \(u \in U\) be any.
There is a \(B_{u, \epsilon} \subseteq M\) such that \(B_{u, \epsilon} \subseteq U\), because \(O\) is induced by \(dist\).
\(\epsilon\) can be taken such that \(\epsilon \lt r\), because if \(r \lt \epsilon\), \(B_{u, r / 2} \subseteq U\) even more, for example.
Then, \(B'_{u, \epsilon} = B_{u, \epsilon} \subseteq U\), by Step 2.
That means that \(U \in O'\).
So, \(O \subseteq O'\).
Step 4:
Let \(U' \in O'\) be any.
Let \(u' \in U'\) be any.
There is a \(B'_{u', \epsilon} \subseteq M\) such that \(B'_{u', \epsilon} \subseteq U'\), because \(O'\) is induced by \(dist'\).
\(\epsilon\) can be taken such that \(\epsilon \lt r\), because if \(r \lt \epsilon\), \(B'_{u, r / 2} \subseteq U'\) even more, for example.
Then, \(B_{u', \epsilon} = B'_{u', \epsilon} \subseteq U'\), by Step 2.
That means that \(U' \in O\).
So, \(O' \subseteq O\).
Step 5:
So, \(O = O'\).
