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proof of properties of the closure operator
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(Proof)
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Recall that the closure of a set in a topological space is defined to be the intersection of all closed sets containing it.
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- : By definition
but since for every
we have
, we immediately find
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is closed
- : Recall that the intersection of any number of closed sets is closed, so the closure is itself closed.
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,
, and

- : If
is any closed set, then
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- : First write down the definition:
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- :
so we have
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where is the set of all limit points of 
- : Let
be a limit point of , and let be a closed set containing . If is not in , then
is an open set containing but not meeting , which implies that
does not meet , which contradicts the fact that was a limit point of . Conversely, suppose that is not a limit point of , and that is not in
. Then there is some open neighborhood of which does not meet . But then
is a closed set containing but not containing , so
.
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"proof of properties of the closure operator" is owned by archibal.
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(view preamble)
Cross-references: neighborhood, meet, implies, open set, limit point, closed, number, closed sets, intersection, topological space, closure
This is version 1 of proof of properties of the closure operator, born on 2004-02-27.
Object id is 5638, canonical name is ProofOfPropertiesOfTheClosureOperator.
Accessed 3326 times total.
Classification:
| AMS MSC: | 54A99 (General topology :: Generalities :: Miscellaneous) |
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Pending Errata and Addenda
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