Total Ordering

Definition A $(T, \leq)$ is a if $\forall x,y \in T$, either $x \leq y$ or $y \leq x$....

11/7/2022

Compact

Definition Let $X$ be a . We say $K \subset X$ is if every of $K$ can be reduced...

11/7/2022

A Nonempty Set is Compact in the Order Topology iff it is Tightly Bounded and Complete

Statement Let $(X, \leq)$ be a endowed with the . Then $K \subset X$ is $K$ is...

11/7/2022

Upper Bound

Definition Let $(P, \leq)$ be a and let $A \subset P$. $x \in P$ is an for $A$ if...

11/7/2022

Lower Bound

Definition Let $(P, \leq)$ be a and let $A \subset P$. $x \in P$ is an for $A$ if...

11/7/2022

Order Topology is Hausdorff

Statement Let $(X, \leq)$ be a with the . Then $X$ is . Proof We break into cases on $X$. $|X| = 1$:...

11/7/2022

Compact Sets in Hausdorff Spaces are Closed

Statement Let $X$ be a . Then if $K \subset X$ is , $K$ is . Proof Let $x \in K^{C}$. Because...

11/7/2022

Closed

Definition Suppose $(X, \tau)$ is a . Then $K \subset X$ is if $K^{C}$ is ....

11/7/2022