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I am pondering the following statements about sets, subsets and topologies in $\mathbb R$.

The empty set is a closed subset of $\mathbb R$ regardless of the topology on $\mathbb R$.

Any open interval is an open subset of $\mathbb R$ regardless of the topology on $\mathbb R$.

Any closed interval is a closed subset of $\mathbb R$ regardless of the topology on $\mathbb R$.

A half-open interval of the form $[a, b)$ is neither an open set nor a closed set regardless of the topology on $\mathbb R$. I think this is a false statement but I am unsure about the first 3. I am in an introduction to proofs class and we are touching on topology. I know these are important distinctions to make because my professor keeps commenting how their is still a lot of confusion about these statements.

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  • $\begingroup$ Technically, all of those statements (except the first one) are false. We can put the trivial topology on $\mathbb{R}$, where the only open sets are $\emptyset$ and $\mathbb{R}$. You can Google the "lower limit topology" to do some reading about your last statement. $\endgroup$ – felani Apr 28 '15 at 20:07
  • $\begingroup$ For instance, generate a topology from the intervals $[a,b)$ as open sets. $\endgroup$ – Simon S Apr 28 '15 at 20:15
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The empty set is a closed subset of $\mathbb R$ regardless of the topology on $\mathbb R$.

This is true: The empty set is closed in any topology, since the empty set is the complement of the full space, and one of the requirements of a topology is that the full space is open.

Any open interval is an open subset of $\mathbb R$ regardless of the topology on $\mathbb R$.

False. Take the trivial topology on $\mathbb R$, which is the topology where only $\mathbb R$ and the empty set are open. All open intervals are nonempty, and almost all open intervals are not $\mathbb R$.

Any closed interval is a closed subset of $\mathbb R$ regardless of the topology on $\mathbb R$.

False again. Take again the trivial topology of $\mathbb R$. Since the complement of a closed interval is neither empty nor all of $\mathbb R$, it is not open in the trivial topology, and thus the closed interval is not closed in that topology.

A half-open interval of the form $[a,b)$ is neither an open set nor a closed set regardless of the topology on $\mathbb R$.

False again. Take the discrete topology in $\mathbb R$, that is the topology in which every set is open. Clearly if every set is open, also the half-open interval is. Moreover, in the discrete topology, every set is closed (because the complement is open), and thus the half-open interval also is closed in the discrete topology.

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