J. Munkres. Topology (2013) Chapter II: Topological Spaces and Continuous Functions. 17: Closed Sets and Limit Points

Closed Sets

Definition

Let \(X\) be a topological space and \(C \subseteq X\). Then \(C\) is closed iff \(X - C\) is open.

Theorem 17.1

Let \(X\) be a topological space.

  1. \(\emptyset\) and \(X\) are closed.
  2. The intersection of a nonempty set of closed sets is closed.
  3. The union of two closed sets is closed.

Theorem 17.2

Let \(X\) be a topological space. Let \(Y\) be a subspace of \(X\) and \(A \subseteq Y\). Then \(A\) is closed in \(Y\) iff there exists a closed set \(C\) in \(X\) such that \(A = C \cap Y\).

Theorem 17.3

Let \(X\) be a topological space. Let \(Y\) be a subspace of \(X\) and \(A \subseteq Y\). If \(A\) is closed in \(Y\) and \(Y\) is closed in \(X\) then \(A\) is closed in \(X\).

Closure and Interior of a Set

Definition

Let \(X\) be a topological space. Let \(A \subseteq X\). The interior of \(A\), denoted \(A^\circ\), is the union of all the open sets included in \(A\).

Definition

Let \(X\) be a topological space. Let \(A \subseteq X\). The closure of \(A\), denoted \(\overline{A}\), is the intersection of all the closed sets that include \(A\).

Theorem 17.4

Let \(X\) be a topological space. Let \(Y\) be a subspace of \(X\). Let \(A \subseteq Y\). Let \(\overline{A}\) be the closure of \(A\) in \(X\). Then the closure of \(A\) in \(Y\) is \(\overline{A} \cap Y\).

Theorem 17.5

Let \(X\) be a topological space. Let \(A \subseteq X\) and \(x \in X\).

  1. \(x \in \overline{A}\) iff every open set containing \(x\) intersects \(A\).
  2. Let \(\mathcal{B}\) be a basis for the topology in \(X\). Then \(x \in \overline{A}\) iff, for all \(B \in \mathcal{B}\), if \(x \in B\) then \(B\) intersects \(A\).

Definition

Let \(X\) be a topological space and \(x \in X\). A neighbourhood of \(x\) is an open set that contains \(x\).

Limit Points

Definition

Let \(X\) be a topological space. Let \(A \subseteq X\) and \(x \in X\). Then \(x\) is a limit point of \(A\) iff every neighbourhood of \(x\) intersects \(A\) in a point other than \(x\) itself.

Theorem 17.6

Let \(X\) be a topological space. Let \(A \subseteq X\). Let \(A'\) be the set of limit points of \(A\). Then

\[ \overline{A} = A \cup A' \enspace . \]

Corollary 17.7

A subset of a topological space is closed if and only if it contains all its limit points.

Hausdorff Spaces

Definition

Let \(X\) be a topological space. Let \((a_n)\) be a sequence of points in \(X\) and \(l \in X\). Then we say \((a_n)\) converges to \(l\), and write \(a_n \rightarrow l\) as \(n \rightarrow \infty\), iff for every neighbourhood \(U\) of \(l\), there exists \(N\) such that \(\forall n \geq N. a_n \in U\).

Definition

A topological space \(X\) is Hausdorff iff, for all \(x,y \in X\), if \(x \neq y\) then there exist disjoint open sets \(U\) and \(V\) with \(x \in U\) and \(y \in V\).

Definition

A topological space \(X\) is \(T_1\) iff, for all \(x \in X\), the set \(\{x\}\) is closed.

Theorem 17.8

Every Hausdorff space is \(T_1\).

Example

Any infinite set under the finite complement topology is \(T_1\) but not Hausdorff.

Theorem 17.9

Let \(X\) be a \(T_1\) space. Let \(A \subseteq X\) and \(x \in A\). Then \(x\) is a limit point of \(A\) if and only if every neighbourhood of \(x\) contains infinitely many points of \(A\).

Theorem 17.10

In a Hausdorff space, a sequence has at most one limit.

Theorem 17.11

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

From the Exercises

Definition

Let \(X\) be a topological space and \(A \subseteq X\). The boundary of \(A\) is the set \(\partial A := \overline{A} \cap \overline{X-A}\).

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