# find closure, interior, and derived sets with respect to topologies

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

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 $\Bbb R.$
• 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.
• @Kathryn: You're exactly right on the first two counts, but your reasoning is a bit shaky. (I will expand my answer to give an alternative approach.) The derived set of $A$ is the set of limit points of $A$--those points $x\in\Bbb R$ such that for any neighborhood $U$ of $x,$ $U$ contains a point of $A$ that is distinct from $x$. Put another way, a point $x\in\Bbb R$ is not in the derived set of $A$ if and only if there is a neighborhood $U$ of $x$ such that $U$ contains at most one point of $A$. Points in the derived set of $A$ may or may not be points of $A$. – Cameron Buie Oct 22 '13 at 0:45
The text passage is a discussion about a math problem involving finding the closure, interior, and derived sets of a set of rational numbers in different topologies. It includes hints and explanations for solving the problem.
no_question

while the passage discusses a mathematical problem, it does not explicitly pose a question that requires a numerical or calculative answer. it provides hints and explanations for solving the problem, but the actual questions (e.g., "find the closure, interior, and derived sets...") are not phrased as direct questions in the text.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# find closure, interior, and derived sets with respect to topologies

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

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 $\Bbb R.$
• 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.
• @Kathryn: You're exactly right on the first two counts, but your reasoning is a bit shaky. (I will expand my answer to give an alternative approach.) The derived set of $A$ is the set of limit points of $A$--those points $x\in\Bbb R$ such that for any neighborhood $U$ of $x,$ $U$ contains a point of $A$ that is distinct from $x$. Put another way, a point $x\in\Bbb R$ is not in the derived set of $A$ if and only if there is a neighborhood $U$ of $x$ such that $U$ contains at most one point of $A$. Points in the derived set of $A$ may or may not be points of $A$. – Cameron Buie Oct 22 '13 at 0:45
user
The text passage is a discussion about a math problem involving finding the closure, interior, and derived sets of a set of rational numbers in different topologies. It includes hints and explanations for solving the problem.

TYPE: NO_QUESTION

While the passage discusses a mathematical problem, it does not explicitly pose a question that requires a numerical or calculative answer. It provides hints and explanations for solving the problem, but the actual questions (e.g., "Find the closure, interior, and derived sets...") are not phrased as direct questions in the text.