A description/proof of that pair of elements of open cover of connected topological space is finite-open-sets-sequence-connected via cover elements
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of connected topological space.
- The reader knows a definition of finite-open-sets-sequence-connected pair of open sets.
Target Context
- The reader will have a description and a proof of the proposition that any pair of elements of any open cover of any connected topological space is finite-open-sets-sequence-connected via some elements of the open cover.
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 connected topological space, \(T\), and any open cover, \(S_c = \{U_\alpha\}, \cup_\alpha U_\alpha = T\), any pair of elements, \(U_1, U_2 \in S_c\), are finite-open-sets-sequence-connected via some elements of \(S_c\).
2: Proof
Take the equivalence class, \(S_e = \{U_\beta\} \subseteq S_c\), of the open cover such that each element of the equivalence class is finite-open-sets-sequence-connected with \(U_1\) via some elements of \(S_c\), which is obviously an equivalence class.
If \(S_e\) did not equal \(S_c\), \(S_r := S_c \setminus S_e = \{U_\gamma\}\) would be nonempty. As \(S_e \cup S_r = S_c\) is an open cover, \((\cup_\beta U_\beta) \cup (\cup_\gamma U_\gamma) = T\), but as \(T\) is connected, \((\cup_\beta U_\beta) \cap (\cup_\gamma U_\gamma) \neq \emptyset\) because otherwise, \(T\) would be the disjoint union of open sets, \(\cup_\beta U_\beta\) and \(\cup_\gamma U_\gamma\). So, there would be a point, \(p \in T\), such that \(p \in \cup_\beta U_\beta\) and \(p \in \cup_\gamma U_\gamma\), which means \(p \in U_\beta\) for an \(\beta\) and \(p \in U_\gamma\) for a \(\gamma\), so, \(p \in U_\beta \cap U_\gamma\), which is a contradiction, because as \(U_\gamma\) would share a point with \(U_\beta\), \(U_\gamma\) should be finite-open-sets-sequence-connected with \(U_1\) via some elements of \(S_c\). So, \(S_e\) equals \(S_c\).
So, \(U_2 \in S_e\), and \(U_1\) and \(U_2\) are finite-open-sets-sequence-connected via some elements of \(S_c\).
3: Note
Although not every connected topological space is path-connected, any pair of open sets of any connected topological space is connected by way of open sets.
