description/proof of that for set, intersection of topologies is topology
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topology.
Target Context
- The reader will have a description and a proof of the proposition that for any set, the intersection of any topologies is a topology.
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:
\(S\): \(\in \{\text{ the sets }\}\)
\(\{O_j \vert j \in J\}\): \(O_j \in \{\text{ the topologies for } S\}\), where \(J \in \{\text{ the possibly uncountable index sets }\}\)
\(O\): \(= \cap_{j \in J} O_J\)
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Statements:
\(O \in \{\text{ the topologies for } S\}\)
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2: Proof
Whole Strategy: Step 1: see that \(O\) satisfies the conditions to be a topology.
Step 1:
Let us see that \(O\) satisfies the conditions to be a topology.
1) \(\emptyset \in O\) and \(S \in O\): \(\emptyset \in O_j\) for each \(j \in J\), so, \(\emptyset \in O\); \(S \in O_j\) for each \(j \in J\), so, \(S \in O\).
2) for any \(U_1 \in O\) and any \(U_2 \in O\), \(U_1 \cap U_2 \in O\): \(U_1, U_2 \in O_j\) for each \(j \in J\), so, \(U_1 \cap U_2 \in O_j\) for each \(j \in J\), so, \(U_1 \cap U_2 \in O\).
3) for any \(U_l \in O\) where \(l \in L\) where \(L\) is any index set not necessarily countable, \((\cup_{l \in L} U_L) \in O\): \(U_l \in O_j\) for each \(j \in J\), so, \((\cup_{l \in L} U_l) \in O_j\) for each \(j \in J\), so, \((\cup_{l \in L} U_l) \in O\).
