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The lower limit topology is finer (has more open sets) than the standard topology on the real numbers (which is generated by the open intervals). The reason is that every open interval can be written as a (countably infinite) union of half-open intervals. For any real and , the interval [,) is clopen in (i.e., both open and closed).
While there are many Borel measures μ, the choice of Borel measure that assigns ((,]) = for every half-open interval (,] is sometimes called "the" Borel measure on . This measure turns out to be the restriction to the Borel σ-algebra of the Lebesgue measure λ {\displaystyle \lambda } , which is a complete measure and is defined on the ...
Open sets in are unions of such rectangles. S {\displaystyle \mathbb {S} } is an example of a space that is a product of Lindelöf spaces that is not itself a Lindelöf space. The so-called anti-diagonal Δ = { ( x , − x ) ∣ x ∈ R } {\displaystyle \Delta =\{(x,-x)\mid x\in \mathbb {R} \}} is an uncountable discrete subset of this space ...
In summary, a set of the real numbers is an interval, if and only if it is an open interval, a closed interval, or a half-open interval. [4] [5] A degenerate interval is any set consisting of a single real number (i.e., an interval of the form [a, a]). [6] Some authors include the empty set in this definition.
In other projects Wikidata item ... Half-open may refer to: Half-open file in chess; Half-open vowel, a class of ... Half-open interval, an interval containing only ...
A classical example is to define a content on all half open intervals [,) by setting their content to the length of the intervals, that is, ([,)) =. One can further show that this content is actually σ-additive and thus defines a pre-measure on the semiring of all half-open intervals.
If is endowed with its usual Euclidean topology then the derived set of the half-open interval [,) is the closed interval [,]. Consider R {\displaystyle \mathbb {R} } with the topology (open sets) consisting of the empty set and any subset of R {\displaystyle \mathbb {R} } that contains 1.
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