Partially ordered set
Partial orders in topological spaces

If P {\displaystyle P} is a partially ordered set that has also been given the structure of a topological space, then it is customary to assume that { (a, b): a ≤ b } {\displaystyle \{(a,b):a\leq b\}} is a closed subset of the topological product space P × P. {\displaystyle P\times P.} Under this assumption partial order relations are well behaved at limits in the sense that if lim i → ∞ a i = a, {\displaystyle \lim _{i\to \infty }a_{i}=a,} and lim i → ∞ b i = b, {\displaystyle \lim _{i\to \infty }b_{i}=b,} and for all i, {\displaystyle i,} a i ≤ b i, {\displaystyle a_{i}\leq b_{i},} then a ≤ b. {\displaystyle a\leq b.}
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