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Let $X = \{0,1,2,3,\ldots\}$ (the non-negative integers), let $$B_1 = \{\{n\} : n \in X \text{ and }n > 0\}= \{\{1\}, \{2\}, \{3\},\ldots\}$$ $$B_2 = \{Z \subset X : X \setminus Z = \{1,2,\ldots n\} \text{ for some }n \in \mathbb{N} \}$$

a) Prove that $B$ is a basis for a topology on $X$.

b) Let $T$ be the topology from part 1. Prove that $(X; T )$ is $T_2$.

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    Also, you said what $B_1$ and $B_2$ are, but not $B$. Perhaps $B=B_1\cup B_2$?2012-04-22
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    The space given by $B=B_1\cup B_2$ is homeomorphic to $\{0\}\cup\{\frac1n; n\in\mathbb N$ as the subspace of the real line with the usual topology. (Perhaps this might help you visualize the problem.) Anyway, you just need to write down the definitions and check.2012-04-22
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    Zev Chonoles, sorry I'll edit my post and you are right B=B1 U B22012-04-22

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