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Question: Let X be a non-empty set. Show that the collection $\tau$ of all subsets of X is a topology on X. This is called the discrete topology.

While I have the definition, I am unsure as to how I can kickstart this question since the set of elements in the collection $\tau$ is not given.

Any help is appreciated. Thanks in advance.

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  • $\begingroup$ the question takes "a topology" to mean a collection of "open" subsets of X which satisfy the three standard axioms. So you are required to show that the collection of all subsets of X satisfies theses axioms. I just notice that Paolo has set this out in his answer $\endgroup$ – David Holden Aug 18 '16 at 4:47
  • $\begingroup$ @DavidHolden That makes it clear then. The question could have been more explicit though. Thanks. $\endgroup$ – Mathematicing Aug 18 '16 at 4:48
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1) $\emptyset$ and $X$ are both subsets of $X$.

2) Let $\{A_i: i \in I\}$ be an arbitrary collection of subsets of $X$. Then $\bigcup_i A_i$ is a subset of $X$.

3) Let $\{B_1,\ldots,B_k\}$ be a finite collection of subsets of $X$. Then $B_1 \cap \cdots \cap B_k$ is a subset of $X$.

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