Important ideas that we discussed in this section include the following.
A subset \(C\) of a topological space \(X\) is closed if \(X \setminus C\) is open.
Any intersection of closed sets is closed, while unions of finitely many closed sets are closed.
A sequence in a topological space \(X\) is a function \(f: \Z^+\) to \(X\text{.}\)
A sequence \((x_n)\) in a topological space \(X\) converges to a point \(x\) in \(X\) if for each open set \(O\) containing \(x\text{,}\) there exists a positive integer \(N\) such that \(x_n \in O\) for all \(n \geq N\text{.}\)
If a sequence \((x_n)\) in a topological space \(X\) converges to a point \(x\text{,}\) then \(x\) is a limit of the sequence \((x_n)\text{.}\)
A limit point of a subset \(A\) of a topological space \(X\) is a point \(x \in X\) such that every neighborhood of \(x\) contains a point in \(A\) different from \(x\text{.}\) A subset \(C\) of a topological space \(X\) is closed if and only if \(C\) contains all of its limit points.
A boundary point of a subset \(A\) of a topological space \(X\) is a point \(x \in X\) such that every neighborhood of \(x\) contains a point in \(A\) and a point in \(X \setminus A\text{.}\) The boundary of \(A\) is the set
\begin{equation*}
\Bdry(A) = \{x \in X \mid x \text{ is a boundary point of } A\}\text{.}
\end{equation*}
A subset \(C\) of \(X\) is closed if and only if \(C\) contains its boundary.
A topological space \(X\) is Hausdorff if we can separate distinct points with open sets in the space. That is, if for each pair \(x\text{,}\)\(y\) of distinct points in \(X\text{,}\) there exists open sets \(O_x\) of \(x\) and \(O_y\) of \(y\) such that \(O_x \cap O_y = \emptyset\text{.}\) Hausdorff spaces are important because sequences have unique limits in Hausdorff spaces and single point sets are closed.
Separation axioms tell us what kinds of objects can be separated by open sets.
In a \(T_1\)-space, we can separate two distinct points with one open set. That is, given distinct points \(x\) and \(y\) in a \(T_1\) topological space \(X\text{,}\) there is an open set \(U\) that separates \(y\) from \(x\) in the sense that \(y \in U\) but \(x \notin U\text{.}\)
In a \(T_2\)-space \(X\) we can separate points more distinctly. That is, if \(x\) and \(y\) are different points in \(X\text{,}\) there exist disjoint open sets \(U\) and \(V\) such that \(x \in U\) and \(y \in V\text{.}\)
In a \(T_3\)-space \(X\) we can separate a point from a closed set that does not contain that point. That is, if \(C\) is a closed subset of \(X\) and \(x\) is a point not in \(C\text{,}\) there exists disjoint open sets \(U\) and \(V\) in \(X\) such that \(C \subseteq U\) and \(x \in V\text{.}\)
In a \(T_4\)-space \(X\) we can separate disjoint closed sets. That is, if \(C\) and \(D\) are disjoint closed subsets of \(X\text{,}\) there exist disjoint open sets \(U\) and \(V\) such that \(C \subseteq U\) and \(D \subseteq V\text{.}\)
