derivation of properties of regular open set
For any , is open. This is obvious.
reverses inclusion. This is also obvious.
and . This too is clear.
, because .
is dense in , because .
. To see this, first note that , so that . Similarly, . Take the union of the two inclusions and the result follows.
. This can be verified by direct calculation:
is regular open iff . See the remark at the end of this entry (http://planetmath.org/DerivationOfPropertiesOnInteriorOperation).
If is open, then is regular open.
Since for any set, . This means . Since is open, is closed, so that . The last inclusion becomes . Taking complement again, we have
Since reverses inclusion, we have , which is one of the inclusions. On the other hand, the inclusion (1) above applies to any open set, and because is open, , which is the other inclusion. ∎
If and are regular open, then so is .
Since are regular open, , which is equal to by property 7 above. Since is open, the last expression becomes by property 9, or by property 7 again. ∎
Remark. All of the properties above can be dualized for regular closed sets. If fact, proving a property about regular closedness can be easily accomplished once we have the following:
is regular open iff is regular closed.
Suppose first that is regular open. Then . The converse is proved similarly. ∎
As a corollary, for example, we have: if is closed, then is regular closed.
If is closed, then is open, so that is regular open by property 9 above, which implies that is regular closed by . ∎
|Title||derivation of properties of regular open set|
|Date of creation||2013-03-22 17:59:24|
|Last modified on||2013-03-22 17:59:24|
|Last modified by||CWoo (3771)|