Open set

In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can "wiggle" or "change" any point x in U by a small amount in any direction and still be inside U.

Related Topics:
Topology - Mathematics - Set

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In other words, if x is surrounded only by elements of U; it can't be on the edge of U.

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As a typical example, consider the open interval (0,1) consisting of all real numbers x with 0 < x < 1.

Related Topics:
Interval - Real number

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If you "wiggle" such an x a little bit (but not too much), then the wiggled version will still be a number between 0 and 1.

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Therefore, the interval (0,1) is open.

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However, the interval (0,1] consisting of all numbers x with 0 < x ≤ 1 is not open; if you take x = 1 and wiggle a tiny bit in the positive direction, you will be outside of (0,1].

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Note that whether a given set U is open depends on the surrounding space, the "wiggle room".

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For instance, the set of rational numbers between 0 and 1 (exclusive) is open in the rational numbers, but it is not open in the real numbers.

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Note also that "open" is not the opposite of "closed".

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First, there are sets which are both open and closed (called clopen sets); in R and other connected spaces, only the empty set and the whole space are clopen, while the set of all rational numbers smaller than √2 is clopen in the rationals.

Related Topics:
Clopen set - Connected space - Empty set

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Also, there are sets which are neither open nor closed, such as (0,1] in R.

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