A description/proof of that for complete metric space, closed subspace is complete
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of complete metric space.
- The reader knows a definition of closed set.
Target Context
- The reader will have a description and a proof of the proposition that for any complete metric space, any closed subspace is complete.
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: Description
For any complete metric space, \(T_1\), any closed subspace, \(T_2 \subseteq T_1\), is complete.
2: Proof
Let \(p_1, p_2, ..., \) where \(p_i \in T_2\) be any Cauchy sequence on \(T_2\). It converges on \(T_1\) to a point, \(p \in T_1\). Let us suppose that \(p \notin T_2\). \(p \in T_1 \setminus T_2\) where \(T_1 \setminus T_2\) is open on \(T_1\). There is an open ball, \(B_{p-\epsilon} \subseteq T_1 \setminus T_2\), around \(p\), and \(p_i \notin B_{p-\epsilon}\) for each \(i\), which is a contradiction against \(p\) is a convergent point of the sequence. So, \(p \in T_2\). The sequence converges to \(p\) on \(T_2\), because for any open ball, \(B'_{p-\epsilon} \subseteq T_2\), \(B'_{p-\epsilon} = B_{p-\epsilon} \cap T_2\), and there is an \(i_0\) such that for any \(i\) such that \(i_0 \lt i\), \(p_i \in B_{p-\epsilon} \cap T_2 = B'_{p-\epsilon}\).
