description/proof of that for set, union of topologies is not necessarily 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 a set, the union of some topologies is not necessarily any 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 \(J \in \{\text{ the possibly uncountable index sets }\}\)
\(O\): \(= \cup_{j \in J} O_J\)
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Statements:
not necessarily \(O \in \{\text{ the topologies for } S\}\)
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2: Proof
Whole Strategy: Step 1: see a counterexample.
Step 1:
Let us see a counterexample.
Let \(S = \{0, 1, 2\}\), \(O_1 = \{\emptyset, \{0\}, S\}\), and \(O_2 = \{\emptyset, \{1\}, S\}\).
Then, \(O = \{\emptyset, \{0\}, \{1\}, S\}\) is not any topology, because \(\{0\} \cup \{1\} = \{0, 1\} \notin O\).
