Originally Posted by

**billym** Like, that a closed set contains its limit points and an open set doesn't?

I'm just wondering if (2,8] and (2,4)U[5,8] are the same in terms of being open or closed.

No.

Plato has said this already, but you seem to have missed it: **Some sets are neither open nor closed.** In fact, some sets (e.g. R) are both!

Here are two very straightforward definitions:

A set $\displaystyle S\subseteq\mathbb{R}$ is *closed* if and only if for each accumulation point (or limit point) $\displaystyle x$ we have $\displaystyle x\in S$.

Also:

A set $\displaystyle T\subseteq\mathbb{R}$ is *open* if and only if for each $\displaystyle x\in T$ there is a neighborhood $\displaystyle Q$ of $\displaystyle x$ with $\displaystyle Q\subseteq T$.

You may also benefit from the following reminders:

A set $\displaystyle Q\subseteq\mathbb{R}$ is a *neighborhood* of $\displaystyle x\in\mathbb{R}$ if and only if there is $\displaystyle \epsilon>0$ with $\displaystyle (x-\epsilon,x+\epsilon)\subseteq Q$.

Also:

If $\displaystyle S\subseteq\mathbb{R}$, then $\displaystyle x$ is an *accumulation point* (or limit point) of $\displaystyle S$ if and only if $\displaystyle x\in\mathbb{R}$ with every neighborhood of $\displaystyle x$ containing infinitely many elements of $\displaystyle S$.

*Please note that this last definition is not necessarily correct outside the context of real analysis. In other branches you must use a more rigorous definition, but for our purposes the above will suffice.*

Okay, so now you are armed with these four definitions. Use them carefully.

Consider for example $\displaystyle \mathbb{R}$. It is trivial that $\displaystyle \mathbb{R}\subseteq\mathbb{R}$. Furthermore, since accumulation points of $\displaystyle \mathbb{R}$ are in $\displaystyle \mathbb{R}$ by definition, we know that $\displaystyle \mathbb{R}$ is a closed set.

But wait!

Notice also that for any accumulation point $\displaystyle x\in\mathbb{R}$ and $\displaystyle \epsilon>0$, we have $\displaystyle (x-\epsilon,x+\epsilon)\subset\mathbb{R}$. So $\displaystyle \mathbb{R}$ is an open set, too!

Thus we have an example of a set which is both open and closed. You can also find examples of a set which is neither open nor closed (e.g. $\displaystyle (0,1]$).

Hopefully that helps.