# Finding interior and closure of specific subsets of $X = \{ a,b,c \}$ with topology $\mathcal{T} = \{X, \emptyset, \{a\}, \{a,b\} \}$

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.

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 $$\{b,c\}$$.