# [SOLVED]Interior of a set

#### topsquark

##### Well-known member
MHB Math Helper
Here's the problem:
Let X be the plane with the usual topology while A = E = the x-axis. Then $$\displaystyle Int_A E = A$$ while $$\displaystyle Int_X E = \emptyset$$, so that the former cannot be obtained by intersecting the latter with A. It is always true, however, that $$\displaystyle Int_A E \supset A \cap Int_X E$$
I'm taking all of this to be that $$\displaystyle E \subseteq A \subseteq X$$ and that X is simply $$\displaystyle \mathbb{R} ^2$$ (with it's usual topology.)

I understand the final equation in the above. I haven't proven it in general but it easily satisfies the mental image that I have. (I probably should make sure to prove it, but this has little to do with the question I'm going to ask.)

So, the question I'm going to ask... Why is $$\displaystyle Int_X E = \varnothing$$? Every open interval in E can be found to be the intersection of an open set in X and E.

Note: My text defines
If X is a topological space and $$\displaystyle E \subset X$$ then $$\displaystyle Int _X E = \cup \{G \in X | \text{G is open and } G \subset E \}$$
Thanks!

-Dan

#### Olinguito

##### Well-known member
Hi Dan.

Every open interval in E can be found to be the intersection of an open set in X and E.
That is true but is nothing to do with the definition of the interior of a set. $\mathrm{Int}_XE$ is the set of all open sets of $X$ contained in $E$; as all nonempty open sests of $X$ contain points outside the $x$-axis, there can be no nonempty open sets of $X$ contained in $E$.

#### topsquark

##### Well-known member
MHB Math Helper
Hi Dan.

That is true but is nothing to do with the definition of the interior of a set. $\mathrm{Int}_XE$ is the set of all open sets of $X$ contained in $E$; as all nonempty open sests of $X$ contain points outside the $x$-axis, there can be no nonempty open sets of $X$ contained in $E$.
Okay. Now that I think about it again I'm thinking I was trying to look at the converse.

Thanks!

-Dan

#### HallsofIvy

##### Well-known member
MHB Math Helper
Rather than "set of all open sets" better wording would be "union of all open sets". The interior of a set is a single set, not a collection of sets.