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.

  • $\begingroup$ 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

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 $X$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.