A bit more topology ... for now
From Vectorcalcumb
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=Compact Sets= | =Compact Sets= | ||
==Speaking Loosely== | ==Speaking Loosely== | ||
| - | *A compact set | + | *A compact set in <m>\\bbR</m> is any finite union of disjoint closed intervals |
*<m> A \\subset \\bbR^n</m> is compact <m>\\leftright~A</m> is closed and bounded. | *<m> A \\subset \\bbR^n</m> is compact <m>\\leftright~A</m> is closed and bounded. | ||
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==Speaking Tightly :)== | ==Speaking Tightly :)== | ||
===General Definition of Compactness=== | ===General Definition of Compactness=== | ||
Revision as of 18:47, 11 February 2006
Contents |
Compact Sets
Speaking Loosely
- A compact set in <m>\\bbR</m> is any finite union of disjoint closed intervals
- <m> A \\subset \\bbR^n</m> is compact <m>\\leftright~A</m> is closed and bounded.
Speaking Tightly :)
General Definition of Compactness
A set <m>A</m> is compact <m>\\leftright</m> every open "covering" of <m>A</m> contains a finite "sub-covering". i.e
<m> A \\subset bigcup{i \\in I}{}{U_i} </m> if <m>A</m> is not closed.
