Given a topological space \(X\) and a set \(A \subseteq X\), we say that a point \(x_0 \in X\) is a limit point of \(A\) if for every open neighborhood \(U \ni x_0\) there is a point \(a \in U \cap A\) with \(a \neq x_0\). We define the closure of \(A\) to be the set \(\overline{A} := A \cup A'\) where \(A'\) is the set of limit points of \(A\). We note that by definition we always have \(A \subseteq \overline{A}\) but not every point in \(A\) is a limit point of \(A\).

We did some examples:

  1. If \(X = \mathbb{R}\) with the standard topology, and \(A = (0, 1)\), then the set of limit points of \(A\) is the set \([0, 1]\) and \(\overline{A} = [0, 1]\).

  2. If \(X = \mathbb{R}\) with the standard topology and \(A = \{\frac{1}{n} \mid n \in \mathbb{Z}^{\ge 1}\}\) then the only limit point of \(A\) is 0 and \(\overline{A} = A \cup \{0\}\).

  3. If \(X = \mathbb{R}\) with the upper topology and \(A = (0, 1)\) then the set of limit points of \(A\) is \([0, \infty)\) and \(\overline{A} = [0, \infty)\).

We say \(A \subseteq X\) is dense if \(\overline{A} = X\). Equivalently, we say \(A\) is dense if for every non-empty open set \(U \subseteq X\) we have \(U \cap A \neq \emptyset\).

Everyone already knows that \(\mathbb{Q}\) is dense in \(\mathbb{R}\) with respect to the standard topology.

We covered three other examples of density when the notion of being dense is equivalent to something very simple. For each topological space \(X\) below, describe exactly when a set \(A \subseteq X\) is dense in \(X\).

  1. \(X = \mathbb{R}\) with the upper topology. We determined that a set \(A \subseteq X\) is dense if and only if it doesn’t have a lower bound.

  2. \(X = \mathbb{R}\) with the discrete topology. We determined that \(A = X\) itself is the only dense set.

  3. \(X = \mathbb{R}\) with the cofinite topology. We did not cover this and left it as something to think about for next week.

We defined the closed sets. A set \(A \subseteq X\) is closed if \(\overline{A} = X\). We proved that equivalently, \(A\) is closed if and only if \(X \setminus A\) is open.

In some final remarks, we mentioned that “closed” does not mean “not open”. In particular, a set can be both open and closed (sometimes called “clopen”), and a set can be neither open nor closed. Note that for any topological space \(X\), the sets \(\emptyset\) and \(X\) are clopen (because they are both defined to be open and are complements of each other).