description/proof of that compact Hausdorff topological space is normal
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of compact topological space.
- The reader knows a definition of Hausdorff topological space.
- The reader knows a definition of normal topological space.
- The reader admits the proposition that any closed subset of any compact topological space is compact.
Target Context
- The reader will have a description and a proof of the proposition that any compact Hausdorff topological space is normal.
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:
\(T\): \(\in \{\text{ the compact Hausdorff topological spaces }\}\)
//
Statements:
\(T \in \{\text{ the normal topological spaces }\}\)
//
2: Proof
Whole Strategy: Step 1: take any closed \(C_1, C_2\) such that \(C_1 \cap C_2 = \emptyset\); Step 2: for each \(c_1 \in C_1\), for each \(c_2 \in C_2\), take an open neighborhood of \(c_1\), \(U_{c_1, c_2}\), and an open neighborhood of \(c_2\), \(U_{c_2}\), such that \(U_{c_1, c_2} \cap U_{c_2} = \emptyset\), take a finite \(\{U_{c_{2, j}}\}\) that covers \(C_2\), and take \(U_{c_1} := \cap U_{c_1, c_{2, j}}\) and \(U_{C_2} := \cup U_{c_{2, j}}\); Step 3: take a finite \(\{U_{c_{1, j}}\}\) that covers \(C_1\) and take \(\cup U_{c_{1, j}}\) and \(\cap U_{C_2, c_{1, j}}\).
Step 1:
Let \(C_1, C_2 \subseteq T\) be any closed subsets such that \(C_1 \cap C_2 = \emptyset\).
Step 2:
Let \(c_1 \in C_1\) be any, which is fixed throughout this Step.
Let \(c_2 \in C_2\) be any.
There are an open neighborhood of \(c_1\), \(U_{c_1, c_2}\), and an open neighborhood of \(c_2\), \(U_{c_2}\), such that \(U_{c_1, c_2} \cap U_{c_2} = \emptyset\), because \(T\) is Hausdorff: \(U_{c_1, c_2}\) depends on \(c_2\) (also \(U_{c_2}\) depends on \(c_1\) but as \(c_1\) is fixed in this Step, the dependence is not explicitly denoted).
\(\{U_{c_2} \vert c_2 \in C_2\}\) is an open cover of \(C_2\).
\(C_2\) is a compact subset of \(T\), by the proposition that any closed subset of any compact topological space is compact.
So, there is a finite subcover of \(\{U_{c_2} \vert c_2 \in C_2\}\), \(\{U_{c_{2, j}} \vert j \in J\}\).
Let us take \(U_{c_1} := \cap_{j \in J} U_{c_1, c_{2, j}}\) and \(U_{C_2} := \cup_{j \in J} U_{c_{2, j}}\).
\(U_{c_1}\) is an open neighborhood of \(c_1\), as a finite intersection.
\(U_{C_2}\) is an open neighborhood of \(C_2\).
\(U_{c_1} \cap U_{C_2} = \emptyset\), because for each \(j \in J\), \(U_{c_1} \subseteq U_{c_1, c_{2, j}}\), so, \(U_{c_1} \cap U_{c_{2, j}} = \emptyset\), and so \(U_{c_1} \cap U_{C_2} = \emptyset\).
Step 3:
By Step 2, for each \(c_1 \in C_1\), there are an open neighborhood of \(c_1\), \(U_{c_1}\), and an open neighborhood of \(U_{C_2, c_1}\), such that \(U_{c_1} \cap U_{C_2, c_1} = \emptyset\): \(U_{C_2, c_1}\) depends on \(c_1\).
\(\{U_{c_1} \vert c_1 \in C_1\}\) is an open cover of \(C_1\).
\(C_1\) is a compact subset of \(T\), as before.
So, there is a finite subcover of \(\{U_{c_1} \vert c_1 \in C_1\}\), \(\{U_{c_{1, j}} \vert j \in J\}\).
Let us take \(U_{C_1} := \cup_{j \in J} U_{c_{1, j}}\) and \(U_{C_2} := \cap_{j \in J} U_{C_2, c_{1, j}}\).
\(U_{C_1}\) is an open neighborhood of \(C_1\).
\(U_{C_2}\) is an open neighborhood of \(C_2\), as a finite intersection of open neighborhoods.
\(U_{C_1} \cap U_{C_2} = \emptyset\), because for each \(j \in J\), \(U_{C_2} \subseteq U_{C_2, c_{1, j}}\), so, \(U_{c_{1, j}} \cap U_{C_2} = \emptyset\), and so, \(U_{C_1} \cap U_{C_2} = \emptyset\).
So, \(T\) is normal.
