Section 17: Closed Sets and Limit Points

Let $X$ be a topological space, and $A\subset X$ (we use $A$ as a subspace or just as a set).

Closed sets

A closed set is the complement of an open set.

...an answer to the mathematician’s riddle: "How is a set different from a door?" should be: "A door must be either open or closed, and cannot be both, while a set can be open, or closed, or both, or neither!"

A topology can be defined in terms of closed sets as a collection of closed sets containing the empty set and the whole space, as well as the intersection of any subcollection of sets and the union of any finite subcollection of sets.

A set is closed in $A$ if its complement in $A$ is open in $A$ .

Properties

A set is closed in $A$ iff it equals the intersection of $A$ with some closed set in $X$ .
If $A$ is closed and $B$ is open in $X$ , then $A-B$ is closed and $B-A$ is open in $X$ .
If $A$ and $B$ are closed in $X$ and $Y$ , respectively, then $A\times B$ is closed in $X\times Y$ .
If $A$ is closed in $X$ , and $B$ is closed in $A$ , then $B$ is closed in $X$ .
A locally finite collection of subsets is a collection of subsets such that for every point in the space there is its neighborhood intersecting only finitely many sets in the collection.
- If a collection of closed subsets is locally finite, then their union is closed.

Closure and interior

The closure of $A$ , $ClA$ or $\overline{A}$ , is the intersection of all closed sets containing $A$ .

The interior of $A$ , $IntA$ , is the union of all open sets contained in $A$ .

The boundary of $A$ is $BdA=\overline{A}\cap\overline{X-A}$ .

$x\in\overline{A}$ iff every (open) neighborhood of $x$ (or, equivalently, every basis element containing $x$ ) intersects $A$ .
The closure of $B\subset A$ in $A$ is, in general, different from the closure of $B$ in $X$ . Therefore, $\overline{B}$ is reserved for the closure of $B$ in $X$ , unless it is stated otherwise.
- To express the closure of $B$ in $A$ one can use the following fact: the closure of $B$ in $A$ equals $\overline{B}\cap A$ .
$IntA$ and $BdA$ are disjoint: $BdA=\overline{A}-IntA$ and $\overline{A}=IntA\cup BdA$ .
The boundary is empty iff the set is both open and closed.

(Kuratowski Theorem) Starting with a given set we can obtain no more than 14 distinct sets by taking operations of complementation and closure (in fact, taking the interior is included as a combination of the other two operations).

See Exercise 21.

There are some more general results, like, for example, if we include the operation of taking the boundary then the number of distinct sets is no more than 34.

Properties

$\overline{\cup_{\alpha}A_{\alpha}}\supset\cup_{\alpha}\overline{A}_{\alpha}$ .
- $\overline{\cup_{i=1}^{n}A_{i}}=\cup_{i=1}^{n}\overline{A}_{i}$ .
$\overline{A\cap B}\subset\overline{A}\cap\overline{B}$ .
$\overline{A-B}\supset\overline{A}-\overline{B}$ , moreover, $\overline{A-B}-\overline{B}=\overline{A}-\overline{B}$ .
$\overline{A\times B}=\overline{A}\times\overline{B}$ .

Limit points

$x$ is a limit point of $A$ if every its neighborhood intersects $A$ in a point different from $x$ , in other words, if $x\in\overline{A-\{x\}}$ . The set of all limit points of $A$ is denoted by $A'$ .

$\overline{A}=A\cup A'$ .
- A subset of a topological space is closed iff it contains all its limit points.

Convergence of sequences

A sequence of points $x_{1},x_{2},\ldots$ converges to the point $x\in X$ iff for every neighborhood $U$ of $x$ there is $N\in\mathbb{Z}_{+}$ such that $x_{n}\in U$ for $n\ge N$ , $n\in\mathbb{Z}_{+}$ . In this case, $x$ is the limit of the sequence $x_{n}$ .

For example, in the ordered square, $\{(a+\frac{1}{n},b)\}$ converges to $(a,1)$ , $\{(a-\frac{1}{n},b)\}$ converges to $(a,0)$ .

Hausdorff Spaces

One’s experience with open and closed sets and limit points in the real line and the plane can be misleading when one considers more general topological spaces. For example, in the spaces $\mathbb{R}$ and $\mathbb{R}^{2}$ , each one-point set $\{x_{0}\}$ is closed... But this fact is not true for arbitrary spaces... Similarly, one’s experience with the properties of convergent sequences in $\mathbb{R}$ and $\mathbb{R}^{2}$ can be misleading... In $\mathbb{R}$ and $\mathbb{R}^{2}$ , a sequence cannot converge to more than one point, but in an arbitrary space, it can.

$T_{0}$ axiom. A topological space satisfies the $T_{0}$ axiom if for every pair of distinct points there is a neighborhood of one of these points that does not contain the other point.

$\mathbb{R}_{+\infty}$ is an example of a $T_{0}$ -space: no finite set is closed in $\mathbb{R}_{+\infty}$ , and $\{1,2,...\}$ converges to every point in $\mathbb{R}$ .

$T_{1}$ axiom. A topological space satisfies the $T_{1}$ axiom if for every pair of distinct points each point has a neighborhood that does not contain the other point.

In a $T_{1}$ -space, every one-point set is closed. In fact, it is an equivalent definition.
$\mathbb{R}_{fc}$ is an example of a $T_{1}$ -space: every finite set is closed in $\mathbb{R}_{fc}$ (and finite sets are the only closed sets except for the space itself), but every sequence having infinite number of different points converges to every point.
If $X$ is a $T_{1}$ -space and $A\subset X$ then $x\in A'$ iff every neighborhood of $x$ contains infinitely many points of $A$ .

$T_{2}$ axiom. A topological space satisfies the $T_{2}$ axiom if every two distinct points have disjoint neighborhoods. A $T_{2}$ -space is also called a Hausdorff space.

A topological space $X$ is a Hausdorff space iff $\{(x,x)|x\in X\}$ is closed in $X\times X$ .
In a Hausdorff space, every one-point set is closed. Therefore, a Hausdorff space is also a $T_{1}$ -space.
In a Hausdorff space, every sequence converges to at most one point.

Properties

A subspace of a Hausdorff space is a Hausdorff space.
The product of two Hausdorff spaces is a Hausdorff space.
An ordered set with the order topology is a Hausdorff space.

Subpages

Section 17: Closed Sets and Limit Points

Section 17: Closed Sets and Limit Points

Closed sets

Properties

Closure and interior

Properties

Limit points

Convergence of sequences

Hausdorff Spaces

Properties