regular open set
Let be a topological space. A subset of is called a regular open set if is equal to the interior of the closure of itself:
Clearly, every regular open set is open, and every clopen set is regular open.
Examples. Let be the real line with the usual topology (generated by open intervals).
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is regular open whenever .
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is not regular open for and . The interior of the closure of is .
If we examine the structure of a little more closely, we see that if we define
then
So an alternative definition of a regular open set is an open set such that .
Remarks.
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For any , is always open.
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and .
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and is dense in .
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and .
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It can be shown that if is open, then is regular open. As a result, following from the first property, , being , is regular open for any subset of .
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In addition, if both and are regular open, then is regular open.
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It is not true, however, that the union of two regular open sets is regular open, as illustrated by the second example above.
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It can also be shown that the set of all regular open sets of a topological space forms a Boolean algebra under the following set of operations:
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and ,
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,
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, and
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.
This is an example of a Boolean algebra coming from a collection of subsets of a set that is not formed by the standard set operations union , intersection , and complementation .
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(a)
The definition of a regular open set can be dualized. A closed set in a topological space is called a regular closed set if .
References
- 1 P. Halmos (1970). Lectures on Boolean Algebras, Springer.
- 2 S. Willard (1970). General Topology, Addison-Wesley Publishing Company.
Title | regular open set |
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Canonical name | RegularOpenSet |
Date of creation | 2013-03-22 15:04:03 |
Last modified on | 2013-03-22 15:04:03 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 9 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 06E99 |
Synonym | regularly open |
Synonym | regularly closed |
Synonym | regularly closed set |
Defines | regular open |
Defines | regular closed |