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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.
If both types of brackets are the same, the entire interval may be referred to as closed or open as appropriate. Whenever infinity or negative infinity is used as an endpoint (in the case of intervals on the real number line), it is always considered open and adjoined to a parenthesis.
In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous. A blog post [1] by T. Tao lists several closed graph theorems throughout mathematics.
In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. [1] [2] A function f : X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph. [3]
It is not closed since its complement in is = (,] [,), which is not open; indeed, an open interval contained in cannot contain 1, and it follows that cannot be a union of open intervals. Hence, I {\displaystyle I} is an example of a set that is open but not closed.
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).
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Some sets are neither open nor closed, for instance the half-open interval [,) in the real numbers. The ray [ 1 , + ∞ ) {\displaystyle [1,+\infty )} is closed. The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points and is nowhere dense.