definition of quotient topology on set with respect to map
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
- The reader knows a definition of surjection.
Target Context
- The reader will have a definition of quotient topology on set with respect to map.
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 topology, \(O_T\)
\( S\): \(\in \{\text{ the sets }\}\)
\( f\): \(: T \to S\), \(\in \{\text{ the surjections }\}\)
\(*O_S\): \(\in \{\text{ the topologies for } S\}\)
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Conditions:
\(\forall S^` \subseteq S (S^` \in O_S \iff f^{-1} (S^`) \in O_T)\)
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2: Note
The quotient topology is indeed a topology, because \(f^{-1} (\emptyset) = \emptyset \in O_T\), so, \(\emptyset \subseteq S\) is open; \(f^{-1} (S) = T \in O_T\), so, \(S \subseteq S\) is open; for any possibly uncountable number of open sets, \(\{U_j \in O_S \vert j \in J\}\), \(f^{-1} (\cup_{j \in J} U_j) = \cup_{j \in J} f^{-1} (U_j)\) 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, which is open, so, \(\cup_{j \in J} U_j \subseteq S\) is open; for any finite number of open sets, \(\{U_j \in O_S \vert j \in J\}\), \(f^{-1} (\cap_{j \in J} U_j) = \cap_{j \in J} (f^{-1} (U_j))\) by the proposition that for any map, the map preimage of any intersection of sets is the intersection of the map preimages of the sets, which is open, so, \(\cap_{j \in J} U_j\) is open.
A typical case is that \(\sim\) is any equivalence relation on \(T\), \(T / \sim\) is the quotient set, and \(f: T \to T / \sim\) is the classification map of \(T\) with respect to \(\sim\), and the quotient topology on \(T / \sim\) with respect to \(f\) is called "quotient topology on \(T / \sim\) with respect to \(\sim\)".
