Difference between revisions of "Second-countability is countable product-closed"
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(New page: {{topospace metaproperty satisfaction}} ==Statement== A product of countably many second-countable spaces, given the product topology, is also a second-countable space. ==Relate...) |
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Latest revision as of 16:53, 21 July 2008
This article gives the statement, and possibly proof, of a topological space property satisfying a topological space metaproperty
View all topological space metaproperty satisfactions | View all topological space metaproperty dissatisfactions
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Property "Page" (as page type) with input value "{{{property}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
Property "Page" (as page type) with input value "{{{metaproperty}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
Statement
A product of countably many second-countable spaces, given the product topology, is also a second-countable space.
Related facts
The result is not true if we give the box topology to the product space.
Proof
Given: A countable family of spaces 
, with each 
having a countable basis 
. 
is the product of the s, given the product topology
To prove: is a second-countable space
