Topology: Closure

Suppose Y is a subset of a topological space X. We define cl(Y) to be the “smallest” closed subset containing Y. Its formal definition is as follows.

  • Let Σ be the collection of all closed subsets C\subseteq X containing Y.
  • Note that X\in \Sigma, so Σ is not empty.
  • Thus, we can take define \text{cl}(Y) := \cap_{C\in\Sigma} C.

This is called the closure of Y in X. Note that the notation cl(Y) does not indicate the ambient space X. If there’s any possibility of confusion, we denote it by clX(Y) instead. The closure satisfies the following properties:

  • \text{cl}(Y) \in \Sigma: since an intersection of closed subsets is still closed;
  • if C\in \Sigma, then since cl(Y) is the intersection of many sets, including C, we must have \text{cl}(Y)\subseteq C.

This justifies our calling cl(Y) the smallest closed subset containing Y.

closureAs an immediate property, we have:

Basic Fact. If Y\subseteq Z are subsets of X, then \text{cl}(Y)\subseteq \text{cl}(Z).

Indeed, cl(Z) is a closed subset of X containing Z, and hence Y. Thus, it must contain cl(Y).

Examples

  1. If Y is closed in X, then cl(Y) = Y.
  2. Take the half-open interval Y = [0, 1)\subseteq \mathbf{R}. The closure is [0, 1].
  3. Take the set of rationals YQ in XR. The closure is the whole real line.
  4. Suppose Y = {3, 5, 7} in N*. What is the closure of Y (see here for the definition of N*)? [ Answer: {1, 2, 3, 4, 5, 6, 7}. ]

warning

Warning: in a metric space X, the closed ball N(a, \epsilon)^* := \{x\in X: d(x, a) \le \epsilon\} is a closed subset since the complement (which contains all x such that d(xa) > ε) is open in X.

However, the closure of the open ball N(a, \epsilon) is not the closed ball N(a, \epsilon)^* in general. For a rather pathological example, consider X = [-1, +1] \cup \{2\} and take the open ball Y = N(0, 2). Then Y = [-1, +1] is already closed in X so cl(Y) = Y, while the closed ball N(0, 2)* = X ≠ cl(Y).

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Properties of Closure

The closure of can be characterised as follows.

Proposition 1. We have \text{cl}(Y) = Y\cup Y^{acc} where Yacc is the collection of all points of accumulation of Y.

Proof.

Let’s first prove that C:=Y\cup Y^{acc} is closed in X.

  • If x\in X-C, then in particular x is not a point of accumulation of Y, so there exists an open subset U\subseteq X containing x such that U ∩ Y contains at most x. But x is not in Y, so U\cap Y=\emptyset, i.e. U\subseteq X-Y.
  • Next we wish to show U\subseteq X-Y^{acc}. Suppose on the contrary y\in U\cap Y^{acc}, so in particular y is a point of accumulation of Y. Since U is an open subset of X containing y, we have U ∩ Y containing a point z ≠ y, which contradicts what we just proved. Thus U\subseteq X - (Y\cup Y^{acc}) = X-C, so XC is a union of open subsets and is thus open.

To complete the proof, since C is a closed subset containing Y, it also contains cl(Y). On the other hand, since cl(Y) is closed, \text{cl}(Y) \supseteq \text{cl}(Y)^{acc} \supseteq Y^{acc}. This gives \text{cl}(Y) \supseteq Y\cup Y^{acc} = C, which completes the proof. ♦

Proposition 2. If Y, Z\subseteq X are subsets, then

\text{cl}(Y\cup Z) = \text{cl}(Y) \cup \text{cl}(Z).

Proof.

Since the RHS is closed and contains Y\cup Z, it must contain the LHS also.

Conversely, since Y\subseteq Y\cup Z, we have \text{cl}(Y) \subseteq \text{cl}(Y\cup Z). Similarly, \text{cl}(Z) \subseteq\text{cl}(Y\cup Z) so the LHS contains the RHS. ♦

Unions and Intersections of Closures

By induction, one can prove that:

\text{cl}(Y_1 \cup Y_2 \cup \ldots \cup Y_n) = \text{cl}(Y_1) \cup \text{cl}(Y_2) \cup \ldots \cup \text{cl}(Y_n)

for any subsets Y_1, Y_2, \ldots, Y_n \subseteq X. However, this is as far as we go, as we cannot extend this to the union of infinitely many sets. E.g. if Y_n = [\frac 1 n, 1] in X=R, then since each Y_n\subseteq X is closed, we have \cup_n\text{cl}(Y_n) = \cup_n Y_n=(0, 1] which isn’t closed, so it cannot possibly be \text{cl}(\cup_n Y_n).

warningAlso, the result is not true for intersection. E.g. let YQ and ZRQ be subsets of the real line R. Then Y ∩ Z is empty, and so is cl(Y ∩ Z). On the other hand, cl(Y) ∩ cl(Z) = R ∩ RR. So we have cl(Y) ∩ cl(Z) ≠ cl(Y ∩ Z).

That being said, we do at least have a one-sided inclusions:

\text{cl}(\cap_i Y_i) \subseteq \cap_i \text{cl}(Y_i), \quad \text{cl}(\cup_i Y_i) \supseteq \cup_i \text{cl}(Y_i)

for any collection of subsets Y_i\subseteq X.

  • For the first inclusion: since \cap_i Y_i \subseteq Y_i for each i, we have \text{cl}(\cap_i Y_i)\subseteq \text{cl}(Y_i). Since this holds for every i, we get the desired inclusion.
  • For the second inclusion: since Y_i\subseteq \cup_i Y_i for each i, we have \text{cl}(Y_i) \subseteq \text{cl}(\cup_i Y_i). Since this holds for each i, we’re done. ♦

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Further Properties

Next, the closure of Y in a smaller subspace can be deduced from the closure in a bigger space.

Proposition 3. If Y\subseteq Z \subseteq X are subsets of a topological space X, then the closure of Y in Z is the intersection of Z with the closure of Y in X.

\text{cl}_Z(Y) = Z\cap \text{cl}_X(Y).

Proof

Since the RHS is a closed subset of Z containing Y, it must contain the LHS.

Conversely, since LHS is closed in Z it must be of the form C ∩ Z for some closed subset C of X. Then C is a closed subset of X containing Y and thus must contain clX(Y). So C ∩ Z must contain the RHS. ♦

Finally, closure also respects the product.

Proposition 4. If Y_1 \subseteq X_1 and Y_2\subseteq X_2 are subsets of topological spaces, then:

\text{cl}(Y_1\times Y_2) = \text{cl}(Y_1) \times \text{cl}(Y_2)

where the LHS closure is taken in X_1\times X_2.

Proof.

Since the RHS is closed in X_1 \times X_2, it must contain the LHS. For the reverse inclusion, we prove by contradiction by picking an element (xy) outside the LHS, cl(Y1 × Y2).

  • In particular, (x, y)\not\in (Y_1\times Y_2)^{acc}, so there is an open subset containing (xy) which doesn’t intersect Y1 × Y2.
  • This open subset must contain a basic open set of the form U × V which contains (xy).
  • Note that we still have (U\times V)\cap (Y_1\times Y_2) = \emptyset and so U\cap Y_1 = \emptyset or V\cap Y_2 = \emptyset.
  • Assume the former; then x\not\in Y_1^{acc}. Since x\in U, we also have x\not\in Y_1, thus giving x\not\in Y_1\cup Y_1^{acc} = \text{cl}(Y_1).

So (xy) is outside the RHS also, and we’re done. ♦

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