What's the difference between open and closed sets?

Question

What's the difference between open and closed sets?

Especially with relation to topology - rigorous definitions are appreciated, but just as important is the intuition!

Answer 1

Intuitively speaking, an open set is a set without a border: every element of the set has, in its neighborhood, other elements of the set. If, starting from a point of the open set, you move away a little, you never exit the set.

A closed set is the complement of an open set (i.e. what stays "outside" from the open set).

Note that some set exists, that are neither open nor closed.

Answer 2

A set X is open if for every point p in X, there exists a neighborhood (open ball) N of p such that N is a subset of X. We call the point p of the set X a limit point if every neighborhood of p has another point q which is also in X. The set X is closed if every limit point of X is a point of X. A set can be both open and closed, and such sets are occasionally termed "clopen." Trivial examples of clopen sets are the empty set (since it has no points, both the above definitions are vacuously true) and the set of all real numbers. You can in turn visualize an open set in R as an open interval on the real line, and a closed set as a closed interval on the real line.

Answer 3

The rigorous definition of open and closed sets is fundamental to topology: you define a topology by saying what its open sets are. From this perspective, open and closed sets are axiomatic, like points and lines in geometry. In any case, closed sets are the complements of open sets and vice versa.

The most familiar example of open sets would be open intervals on the real line, intervals of the form {x : a < x < b}. Such sets and their arbitrary unions define the standard topology on the real line.

Note that the paragraph above describes the standard topology, but not the only one. You could put a very different topology on the real line. That's because a topology is determined by what you call open sets and not by the underlying space per se. For example, another topology on the real line defines a set to be open if its complement has only a finite number of points.

Answer 4

An open set is a set S for which, given any of its element A, you can find a ball centered in A and whose points are all in S.

A closed set is a set S for which, if you have a sequence of points in S who tend to a limit point B, B is also in S.

Intuitively, a closed set is a set which contains its own boundary, while an open set is a set where you are able not to leave it if you move just a little bit.

Answer 5

I will not reiterate the very nice definitions found in the other answers, however I think that these "practical" definitions might help you as well on an intuitive level.

Open sets are typically used as domains for functions, as they are more useful for analysing "continuous" properties like differentiability. Also they don't have nasty borders (hence you don't have to deal with functions which are well behaved only on one side of the edge).

Closed sets are useful because, if they are limited, they are compact.

Answer 6

The set $\tau$ of open subsets of a set $X$ is an algebraic structure with $\cup$ and $\cap$. The intersection of two sets in $\tau$ should be a set in $\tau$. And the union $\displaystyle\bigcup_{i\in I}\mathcal O_{i}$ should be in $\tau$ for any set of open sets $\{\mathcal O_{i}\}_{i\in I}$. Also $\emptyset,X\in\tau$. And that's all.

The set $\sigma$ of closed subsets of a set $X$ is the set $\{\complement_X\mathcal O \}_{\mathcal O\in\tau}$ which is the dual structure such that the union of any two closed sets is a closed set and such that $\displaystyle\bigcap_{i\in I}\mathcal F_{i}$ is closed for any set of closed sets $\{\mathcal F_{i}\}_{i\in I}$. And $\emptyset,X$ are both open and closed.

Any of those two structures defines a topology on $X$, which adds a concept of proximity to $X$. With a topology on a set $X$ it is not only possible to decide what elements thats belongs to a subset of $X$, but also which elements in $X$ that is proximate to that subset:

$x\in \overline A \Leftrightarrow \forall\mathcal O\in\tau:x\in\mathcal O\Rightarrow A\cap\mathcal O\neq\emptyset\qquad$ or dually

$\displaystyle\overline A=\bigcap_{\mathcal F\in\sigma_A}\mathcal F$, where $\sigma_A=\{\mathcal F\in\sigma|A\subseteq \mathcal F\}$.

The proximity is an extension of the membership to a set in a topological space $(X,\tau)$ (or dually $(X,\sigma)$) and can also be defined explicit and then define the topology it self:

Given a binary relation $\propto\subseteq X^2$ such that

1. $\neg\exists x\in X:x\propto\emptyset$
2. $x\in A\Rightarrow x\propto A$
3. $x\propto A\subseteq B \Rightarrow x\propto B$
4. $x\propto A\cup B\Rightarrow x\propto A \vee x\propto B$.

Then $\propto$ defines a topological proximity relation (and a topology) on X by $x\in\overline A\Leftrightarrow x\propto A$.

An other old man's thinking about topology