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A space, $X$, is compact if every open cover of $X$ contains a finite subcover.

A space, $X$, is compact if for every collection ${\cal C}$ of closed sets in $X$ having the finite intersection property, $\cap_{C \in {\cal C}} C \not= \emptyset$.


A space, $X$, is limit point compact if every infinite subset of $X$ has a limit point.

A space, $X$, is sequentially compact if every sequence has a convergent subsequence.



Thm 28.1  Compactness implies limit point compactness, but not conversely.

HW: p. 178: 4, 6, p. 194: 1
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HW:  limit point compactness does not implies sequential compactness.
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HW:  Suppose $X$ is first countable and $T_1$, then limit point compactness implies sequential compactness.











Thm 28.2:  In a metrizable space $X$, the following are equivalent

1.) $X$ is compact
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2.) $X$ is limit point compact
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3.)  $X$ is sequentially compact.


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limit point compactness does not implies sequentially compactness.


compactness does not imply sequential compactness (Hint consider $[0, 1]^\omega$).


Sequentially compact implies countably compact, but not vice versa.

HW:  Sequential compactness implies limit point compactness.


2a.) Given the following digraph, D, find the ergodic chains and the absorbing chain that can be used to study D.


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