A bit more topology ... for now
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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=== | ||
A set <m>A</m> is compact <m>\\leftright</m> every open "covering" of | A set <m>A</m> is compact <m>\\leftright</m> every open "covering" of | ||
| - | <m>A</m> contains a finite "sub-covering". i.e <br> <m> A \\subset bigcup{i \\in I}{}{U_i} </m> | + | <m>A</m> contains a finite "sub-covering". i.e <br> if <m> A \\subset bigcup{i \\in I}{}{U_i} </m>, with all sets <m> U_i</m> open sets, then there exists<br> a finite set of indices |
| - | + | <m> J = \\delim{lbrace} i_1,...,i_m \\delim{rbrace} \\subset I</m> such that <m> A \\subset U_{i_1} bigcup{}{} ... bigcup{}{} U_{i_m} </m> | |
Current revision as of 18:57, 11 February 2006
Contents |
[edit] Compact Sets
[edit] 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.
[edit] Speaking Tightly :)
[edit] 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
if <m> A \\subset bigcup{i \\in I}{}{U_i} </m>, with all sets <m> U_i</m> open sets, then there exists
a finite set of indices
<m> J = \\delim{lbrace} i_1,...,i_m \\delim{rbrace} \\subset I</m> such that <m> A \\subset U_{i_1} bigcup{}{} ... bigcup{}{} U_{i_m} </m>
