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Intersection and Union of Intervals

TOPO-N4JWGP

$X, \leq$ is a partially ordered set, $T=\{U\subset X:\forall x\in U, y\in X, x < y\implies y\in U\}$.

Which of the following statements are TRUE?

Select ALL that apply.

A

$\forall S\subset T$, $\cap_{V\in S} V\in T$.

B

$\forall S\subset T$, $\cup_{V\in S} V\in T$.

C

$\subseteq$ is a linear order on $T$.

D

$\forall S\subset T$, there is a infimum of $S$ under $\subseteq$.

E

$\forall S\subset T$, $S$ has a least element under $\subseteq$.