definition of trivial topological space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
Target Context
- The reader will have a definition of trivial topological space.
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:
\(*T\): \(\in \{\text{ the topological spaces }\}\), with the topology specified below, \(O\)
//
Conditions:
\(O = \{\emptyset, T\}\)
//
2: Note
Let us see that \(O\) is indeed a topology.
1) \(\emptyset \in O\) and \(T \in O\).
2) for each \(U_1 \in O\) and each \(U_2 \in O\), \(U_1 \cap U_2 \in O\): when \(U_1 = \emptyset\) or \(U_2 = \emptyset\), \(U_1 \cap U_2 = \emptyset \in O\); when \(U_1 = T\) and \(U_2 = T\), \(U_1 \cap U_2 = T \in O\).
3) for each \(\{U_j \in O \vert j \in J\}\) where \(J\) is any index set not necessarily countable, \((\cup_{j \in J} U_j) \in O\): when \(T \in \{U_j \in O \vert j \in J\}\), \((\cup_{j \in J} U_j) = T \in O\); otherwise, \((\cup_{j \in J} U_j) = \emptyset \in O\).
So, \(O\) is a topology.
