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Topology

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Properties of Directed Sets 3

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Which of the following statements are true?

Select ALL that apply.

A

A totally ordered set is a directed set.

B

An equivalence relation is a directed set if and only if it has only one equivalence class.

C

Let $\preceq_i$, $i=1,2,\dots$ be a sequence of relations on $X$, if $\preceq_i\subseteq\preceq_{i+1}$, and $X$ is a directed set under $\preceq_i$ for all $i$, then $X$ is a directed set under $\cup_i\preceq_i$.

D

Let $\preceq_i$, $i=1,2,\dots$ be a sequence of relations on $X$, if $\preceq_i\supseteq\preceq_{i+1}$, and $X$ is a directed set under $\preceq_i$ for all $i$, then $X$ is a directed set under $\cap_i\preceq_i$.

E

If $(X,\preceq)$ is a directed set, then $\sim$ defined as $a\sim b\iff a\preceq b$ and $b\preceq a$ is an equivalence relation.