So far, we’ve dealt with the motivation and definition of a metric space, and have also seen some standard examples. Pretty soon we’ll be moving onto different ways we can categorise metric spaces, but before we do that we’ll need to deal with a topic that is crucial to any kind of development: Open and closed sets.
The first misconception to do away with is the idea that open and closed sets set up a dichotomy in terms of classification. Once we’ve defined what they are, it might be a little clearer why that’s not the case — for now, just keep in mind that any one particular set can be:
- Open
- Closed
- Open and closed
- Neither open or closed
Also, I’m cheating here a little bit for the sake of simplicity; it’s possible for the open/closed distinction to be perfectly meaningless too. If I want ‘open’ and ‘closed’ to have any kind of meaning, we need to define which sets are open (under the restriction of some basic rules). If you’ve ever heard of topology, it’s exactly that: A topology for a particular space defines what the open sets are in that space.
Going through all the background for that is a bit fiddly, however, with not much payoff for what we’re interested in. So for the moment, we’ll work with the understanding that when I talk about open sets, I’m specifically talking about open sets in a metric space.
With that out of the way, the first thing we’ll need to see are open balls. The open ball of radius 
centered at the point 
(where 
is our set) is
which can be read as “the set of points in of distance less than away from 
” . So, for example, the open ball 
in 
is just the interval 
.
We call an element of a set 
an interior point if we can find an open ball centred on that point which is entirely contained in . I.e., 
is an interior point if and only if for some 
,
.
An open set is a set in which every point is an interior point. So for example, in the set 
is open, since for any given point I can always choose an interval that stays contained in 
— e.g. 
is interior, which I can show by choosing the open ball 
.
Now, imagine that you have some sequence of points 
in a set 
and suppose that they converge to some point that we shall call . Then is called an adherent point of 
.
A set is called closed if it contains all of its adherent points — in other words, any convergent sequence of points that lives entirely in must have its limit also in .
An example of a closed set in is the interval ![[0,1] = \{x : 0 \leq x \leq 1\}](http://s.wordpress.com/latex.php?latex=%5B0%2C1%5D%20%3D%20%5C%7Bx%20%3A%200%20%5Cleq%20x%20%5Cleq%201%5C%7D%20&bg=ffffff&fg=000000&s=0 "[0,1] = \{x : 0 \leq x \leq 1\}")
. Showing this is true is slightly more work than showing that is open (though quite easy if you have a few other simple tricks in your mathematical arsenal); however I invite people to play around constructing sequences in that interval, and noting that they all must converge in the interval.
Now, just to finish up: We’ve seen an example of an open set, and one of a closed set; so what of the others categories? For a set that is neither open nor closed, we turn to the interval 
. It is clearly not open — 
, but any open ball around 0 will necessarily contain negative numbers which are not in our original set. It is also not closed, which we can see by examining the sequence
.
Every element of that sequence is found in our interval (i.e. 
), yet the limit as 
is clearly 
.
Finally, for a set that is both open and closed, we turn to the set itself. By construction, every convergent sequence in converges to a point in (the details of that are unimportant for this discussion — though it should, at least, seem quite true); and if I choose any point in and place around it an open ball of any radius, it will still be contained in .
Somewhat paradoxically, another example of a set that is both open and closed is the empty set which contains no elements, 
. It fulfills all of the criteria by dint of it being completely trivial — for instance it is technically true that every convergent sequence in the empty set has a limit in the empty set, even if this is only achieved by having no sequences to begin with. The sneaky little bugger.
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and two other sides 
and 
it is true that
.