Classnotes 051115

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An example of a limit-point compact topological space that it is not compact: Let $\mathbf{X}=\mathbb{Z}$ and define $\mathbf{B}_n=\{2n-1,2n\}$ for each $n$ . Obviously the collection of all $\mathbf{B}_n$ is a basis for a topology on $\mathbf{X}$ . Notice that with this topology $\mathbf{X}$ is not compact: the same family of all $\mathbf{B}_n$ is an open cover of $\mathbf{X}$ with no finite subcover. But $\mathbf{X}$ is limit-point compact: suppose that $\mathbf{A}$ is an infinite subset of $\mathbf{X}$ and $a\in \mathbf{A}$ . Define $x = 2 n - 1$ if $a$ is even, and $x = 2 n$ if $a$ is odd. Then every open set that contains $x$ also contains $a$ , so $x$ is an accumulation point of $\mathbf{A}$ .

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Classnotes 051115

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