description/proof of that closed subspace of paracompact topological space is paracompact
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of paracompact topological space.
- The reader knows a definition of topological subspace.
- The reader knows a definition of closed subset of topological space.
- The reader admits the proposition that for any topological space and any point on any subspace, the intersection of any neighborhood of the point on the base space and the subspace is a neighborhood on the subspace.
Target Context
- The reader will have a description and a proof of the proposition that any closed subspace of any paracompact topological space is paracompact.
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:
\(T'\): \(= \{\text{ the paracompact topological spaces }\}\)
\(T\): \(\in \{\text{ the closed subspaces of } T'\}\)
//
Statements:
\(T \in \{\text{ the paracompact topological spaces }\}\)
//
2: Proof
Whole Strategy: Step 1: for each open cover of \(T\), \(\{U_j \vert j \in J\}\), take the open cover of \(T'\), \(\{U'_J \vert j \in J\} \cup \{T' \setminus T\}\), where \(U_j = U'_j \cap T\); Step 2: take any locally finite refinement of \(\{U'_J \vert j \in J\} \cup \{T' \setminus T\}\), \(\{V'_l \vert l \in L\}\), and see that \(\{V'_l \cap T \vert l \in L\} \setminus \{\emptyset\}\) is a locally finite refinement of \(\{U_j \vert j \in J\}\).
Step 1:
Let \(\{U_j \vert j \in J\}\) where \(J\) is any possibly uncountable index set be any open cover of \(T\).
For each \(j \in J\), \(U_j = U'_j \cap T\) where \(U'_j \subseteq T'\) is an open subset, by the definition of topological subspace.
Let us take \(\{U'_J \vert j \in J\} \cup \{T' \setminus T\}\), which is an open cover of \(T'\), because \(U'_J \subseteq T'\) is open and \(T' \setminus T \subseteq T'\) is open, and for each \(t' \in T'\), \(t' \in T\) or \(t' \in T' \setminus T\), and when \(t' \in T\), \(t' \in U_j\) for a \(j \in J\), so, \(t' \in U'_j \cap T \subseteq U'_j\), and when \(t' \in T' \setminus T\), \(t' \in T' \setminus T\).
Step 2:
As \(T'\) is paracompact, there is a locally finite refinement of \(\{U'_J \vert j \in J\} \cup \{T' \setminus T\}\), \(\{V'_l \vert l \in L\}\).
Let us take \(\{V'_l \cap T \vert l \in L\} \setminus \{\emptyset\} = \{V'_l \cap T \vert l \in L^`\}\) where \(L^` \subseteq L\): in fact, \(\emptyset\) does not need to be removed although it is useless.
It is an open cover of \(T\), because \(V'_l \cap T\) is open on \(T\), by the definition of topological subspace, and for each \(t \in T\), as \(t \in T'\), \(t \in V'_l\) for an \(l \in L\), and \(t \in V'_l \cap T\), while \(l \in L^`\), because \(V'_l \cap T \neq \emptyset\).
For each \(l \in L^`\), \(V'_l \subseteq U'_j\) for a \(j \in J\) (\(V'_l \subseteq T' \setminus T\) is impossible, because otherwise, \((V'_l \cap T) \subseteq ((T' \setminus T) \cap T) = \emptyset\)), so, \(V'_l \cap T \subseteq U'_j \cap T = U_j\), so, \(\{V'_l \cap T \vert l \in L^`\}\) is a refinement of \(\{U_j \vert j \in J\}\).
Let us see that \(\{V'_l \cap T \vert l \in L^`\}\) is locally finite on \(T\).
Let \(t \in T\) be any.
\(t \in T'\), so, there is a neighborhood of \(t\), \(N'_t \subseteq T'\), such that \(N'_t\) intersects only some finite \(\{V'_{l_1}, ..., V'_{l_n}\}\) among \(\{V'_l \vert l \in L\}\), because \(\{V'_l \vert l \in L\}\) is locally finite.
\(N'_t \cap T \subseteq T\) is a neighborhood of \(t\), by the proposition that for any topological space and any point on any subspace, the intersection of any neighborhood of the point on the base space and the subspace is a neighborhood on the subspace.
Then, \(N'_t \cap T\) can intersect only \(\{V'_{l_1} \cap T, ..., V'_{l_n} \cap T\}\) among \(\{V'_l \cap T \vert l \in L\}\), because when \(N'_t \cap V'_l = \emptyset\), \(N'_t \cap V'_l \cap T = \emptyset \cap T = \emptyset\), while \(N'_t \cap V'_l \cap T = N'_t \cap V'_l \cap T \cap T = (N'_t \cap T) \cap (V'_l \cap T)\), then, \(N'_t \cap T\) can intersect only \(\{V'_{l_1} \cap T, ..., V'_{l_n} \cap T\} \setminus \{\emptyset\}\) among \(\{V'_l \cap T \vert l \in L^`\}\).
So, \(\{V'_l \cap T \vert l \in L^`\}\) is locally finite on \(T\).
So, \(T\) is paracompact.
3: Note
\(T \subseteq T'\) needs to be closed, because otherwise, we could not construct an open cover of \(T'\) as \(\{U'_J \vert j \in J\} \cup \{T' \setminus T\}\), and we would need to construct like \(\{U'_J \vert j \in J\} \cup \{U'_m \vert m \in M\}\), but taking a locally finite refinement of it, \(\{V'_l \vert l \in L\}\), \(\{V'_l \cap T \vert l \in L\}\) would not be guaranteed to be a refinement of \(\{U_j \vert j \in J\}\), because it might be \(V'_l \cap T \subseteq V'_l \subseteq U'_m\) for an \(m \in M\), and \(V'_l \cap T \subseteq U'_m \cap T\) would not imply \(U'_m \cap T \in \{U_j \vert j \in J\}\).
