description/proof of that in order to check continuousness of map, preimages of only basis or subbasis are enough
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of continuous map.
- The reader knows a definition of basis of topological space.
- The reader knows a definition of subbasis of topological space.
- The reader admits some criteria for any collection of open sets to be a basis.
- The reader admits the proposition that for any map, the map preimage of any union of sets is the union of the map preimages of the sets.
- The reader admits the proposition that for any map, the map preimage of any intersection of sets is the intersection of the map preimages of the sets.
Target Context
- The reader will have a description and a proof of the proposition that in order to check the continuousness of any map between any topological spaces, the preimages of only any basis or any subbasis are enough to be checked.
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:
\(T_1\): \(\in \{\text{ the topological spaces }\}\)
\(T_2\): \(\in \{\text{ the topological spaces }\}\)
\(f\): \(: T_1 \to T_2\)
\(B_2\): \(\in \{\text{ the bases of } T_2\}\)
\(B'_2\): \(\in \{\text{ the subbases of } T_2\}\)
//
Statements:
(
\(\forall U_\beta \in B_2 (f^{-1} (U_\beta) \in \{\text{ the open subsets of } T_1\})\)
\(\implies\)
\(f \in \{\text{ the continuous maps }\}\)
)
\(\land\)
(
\(\forall U_\beta \in B'_2 (f^{-1} (U_\beta) \in \{\text{ the open subsets of } T_1\})\)
\(\implies\)
\(f \in \{\text{ the continuous maps }\}\)
)
//
2: Proof
Whole Strategy: Step 1: express each open subset, \(U_2 \subseteq T_2\), as the union of some elements of \(B_2\) and express its preimage under \(f\) as the union of the preimages of the elements of \(B_2\); Step 2: express each open subset, \(U_2 \subseteq T_2\), as the union of some intersections of some finite elements of \(B'_2\) and express its preimage under \(f\) as the union of the intersections of the finite preimages of the elements of \(B'_2\).
Step 1:
Any open subset, \(U_2 \subseteq T_2\), is \(U_2 = \cup_{\beta \in B} U_\beta\) where \(B\) is a possibly uncountable index set and \(U_\beta \in B_2\), by some criteria for any collection of open sets to be a basis.
\(f^{-1} (U_2) = \cup_{\beta \in B} f^{-1} (U_\beta)\), by the proposition that for any map, the map preimage of any union of sets is the union of the map preimages of the sets.
If \(f^{-1} (U_\beta)\) is open on \(T_1\), \(\cup_{\beta \in B} f^{-1} (U_\beta)\) is open on \(T_1\) as a union of open subsets.
Step 2:
\(U_2 \subseteq T_2\), is \(U_2 = \cup_{\beta \in B} \cap_{j \in J_\beta} U'_{\beta, j}\) where \(B\) is a possibly uncountable index set, \(J_\beta\) is a finite index set for each \(\beta\), and \(U'_{\beta, j} \in B'_2\), by some criteria for any collection of open sets to be a basis.
\(f^{-1} (U_2) = \cup_{\beta \in B} f^{-1} (\cap_{j \in J_\beta} U'_{\beta, j}) = \cup_{\beta \in B} (\cap_{j \in J_\beta} f^{-1} (U'_{\beta, j}))\), by the proposition that for any map, the map preimage of any union of sets is the union of the map preimages of the sets and the proposition that for any map, the map preimage of any intersection of sets is the intersection of the map preimages of the sets.
If \(f^{-1} (U'_{\beta, j})\) is open on \(T_1\), \(\cap_{j \in J_\beta} f^{-1} (U'_{\beta, j})\) is open on \(T_1\) as a finite intersection of open subsets, and \(\cup_{\beta \in B} (\cap_{j \in J_\beta} f^{-1} (U'_{\beta, j}))\) is open on \(T_1\) as a union of open subsets.
