Say I have a set $X = \{ a,b,c \}$ with topology $\mathcal{T} = \{X, \emptyset, \{a\}, \{a,b\} \}$.
If I'm trying to figure out the interior and closure of a particular subset $Y$, then what elements of $X$ are contained in $Y$ matter immensely.
I know the interior of $Y$ is the union of all open sets contained in $Y$. The union of open sets is again an open set. Hence the interior of $Y$ is the largest open set contained in $Y$.
The closure of a subset $Y$ of points in a topological space is the intersection of all closed sets containing $Y$.
Let $X$ is as above with topology $\mathcal{T}$. If $Y = \{a\}$ or $Y = \{b\}$, then I'm trying to find out what the interior and closure of $Y$ are.
There is no set containing $b$ and a subset of $Y$, so $b$ must not be an interior point of $Y$. We see $\{a\}$ is an open set containing $a$ and is a subset of $Y$, but does $\{a,b\}$ factor in as well?
The problem I'm having is figuring out how the elements of the subset affect finding the interior and closure in the context of $Y = \{a\}$ and $Y = \{b\}$. For $Y = \{a\}$, I would guess that the interior is just $\{a\}$, but I'm not entirely certain.
For the closures specifically, I'm having trouble identifying the closed sets to use.
Best Answer
If $Y=\{a\}$, then $Y\in\mathcal T$, and therefore the interior of $Y$ is $Y$ itself. On the other hand, both $b$ and $c$ are limit points of $Y$, and therefore the closure of $Y$ is $X$.
If $Y=\{b\}$, then then only open subset of $Y$ is $\emptyset$, and therefore the interior of $Y$ is $\emptyset$. And $a$ is not a limit point of $Y$, since $a\in\{a\}$ and $\{a\}$ is a neighborhood of $a$ which contains no point of $Y$. But $c$ is a limit point of $Y$ (the only neighborhood of $c$ is $X$). So, the closure of $Y$ is $\{b,c\}$.