[Math] find closure, interior, and derived sets with respect to topologies

general-topologyreal-analysis

I have a midterm coming up and I'm trying to understand this problem. I understand what closure and interior mean but the different topologies are a little confusing to me.

Let $\mathbb{R}$ be the set of real numbers and $A= \{ x: x \mathrm{\, is \, rational \}}$.

Find the closure, interior, and derived sets of $A$ with respect to the discrete topology, the indiscrete topology and the topology formed by defining a set to be open if it contains all but at most countably many points

Best Answer

Hints: In the discrete topology, every subset of $\Bbb R$ is both closed and open. In the indiscrete topology, only the empty set and all of $\Bbb R$ are open (or closed). In the cocountable topology (the typical name for that last one), note that the closed subsets of $\Bbb R$ are $\Bbb R$ itself, and every at most countable subset of $\Bbb R.$

A few useful results that may help you with this problem (and are good exercises to prove) are the following:

  • The closure of $A$ is the smallest closed set $C$ such that $A\subseteq C$. In particular, then, $A$ is closed if and only if it is equal to its closure.
  • The interior of $A$ is the largest open set $U$ such that $U\subseteq A.$ In particular, then, $A$ is open if and only if it is equal to its interior.
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