definition of cross section of map from subset of product set by element of subproduct set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of product set.
- The reader knows a definition of map.
- The reader knows a definition of cross section of subset of product set by element of subproduct set.
Target Context
- The reader will have a definition of cross section of map from subset of product set by element of subproduct set.
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:
\( J'\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\( \{S_{j'} \in \{\text{ the sets }\} \vert j' \in J'\}\):
\( \times_{j' \in J'} S_{j'}\): \(= \text{ the product set }\)
\( S\): \(\subseteq \times_{j' \in J'} S_{j'}\)
\( S'\): \(\in \{\text{ the sets }\}\)
\( f\): \(: S \to S'\)
\( J\): \(\subset J'\), such that \(J \neq \emptyset\)
\( \times_{j \in J} s_j\): \(\in \times_{j \in J} S_j\)
\( S_{[\times_{j \in J} s_j]}\): \(= \text{ the cross section }\)
\(*f_{[\times_{j \in J} s_j]}\): \(: S_{[\times_{j \in J} s_j]} \to S', \times_{l \in J' \setminus J} s_l \mapsto f (\times_{j' \in J'} s_{j'})\)
//
Conditions:
//
2: Note
\(S_{[\times_{j \in J} s_j]}\) may be empty.
