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The empty set and the set of all reals are both open and closed intervals, while the set of non-negative reals, is a closed interval that is right-open but not left-open. The open intervals are open sets of the real line in its standard topology, and form a base of the open sets. An interval is said to be left-closed if it has a minimum element ...
The open sets of the usual Euclidean topology of the real line are the empty set, the open intervals and every union of open intervals. The interval I = ( 0 , 1 ) {\displaystyle I=(0,1)} is open in R {\displaystyle \mathbb {R} } by definition of the Euclidean topology.
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).
The open interval (0,1) is the set of all real numbers between 0 and 1; but not including either 0 or 1. To give the set (0,1) a topology means to say which subsets of (0,1) are "open", and to do so in a way that the following axioms are met: [1] The union of open sets is an open set. The finite intersection of open sets is an open set.
Though the subspace topology of Y = {−1} ∪ {1/n } n∈N in the section above is shown not to be generated by the induced order on Y, it is nonetheless an order topology on Y; indeed, in the subspace topology every point is isolated (i.e., singleton {y} is open in Y for every y in Y), so the subspace topology is the discrete topology on Y (the topology in which every subset of Y is open ...
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If has its usual Euclidean topology then the open set = (,) (,) is not a regular open set, since (¯) = (,). Every open interval in R {\displaystyle \mathbb {R} } is a regular open set and every non-degenerate closed interval (that is, a closed interval containing at least two distinct points) is a regular closed set.
The set Γ of all open intervals in forms a basis for the Euclidean topology on .. A non-empty family of subsets of a set X that is closed under finite intersections of two or more sets, which is called a π-system on X, is necessarily a base for a topology on X if and only if it covers X.