Examples
The irreducibility depends much on actual topology on some set. For example, possibly contradicting the intuition, the real numbers (with their usual topology) are reducible: for example the open interval (−1, 1) is not dense, its closure is the closed interval .
However, the notion is fundamental and more meaningful in algebraic geometry: consider the variety
- X := {x · y = 0}
(a subset of the affine plane, x and y are the variables) endowed with the Zariski topology. It is reducible, its irreducible components are its closed subsets {x = 0} and {y = 0}.
This can also be read off the coordinate ring k/(xy) (if the variety is defined over a field k), whose minimal prime ideals are (x) and (y).
This article incorporates material from irreducible on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. This article incorporates material from Irreducible component on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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