# Centered set

In mathematics, in the area of order theory, an **upwards centered set** *S* is a subset of a partially ordered set, *P*, such that any finite subset of *S* has an upper bound in *P*. Similarly, any finite subset of a **downwards centered set** has a lower bound. An upwards centered set can also be called a **consistent set**. Note that any directed set is necessarily centered, and any centered set is linked.

A subset *B* of a partial order is said to be **σ-centered** if it is a countable union of centered sets.

## References

- Fremlin, David H. (1984).
*Consequences of Martin's axiom*. Cambridge tracts in mathematics, no. 84. Cambridge: Cambridge University Press. ISBN 0-521-25091-9. - Davey, B. A.; Priestley, Hilary A. (2002), "9.1",
*Introduction to Lattices and Order*(2nd ed.), Cambridge University Press, p. 201, ISBN 978-0-521-78451-1, Zbl 1002.06001.

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