General topology begins by declaring which subsets are observable as open. From that single primitive one recovers convergence, closure, continuity, compactness, and separation.
Topologies on a set
The axioms isolate exactly the set operations that open balls satisfy in metric spaces.
Definition. A topology on a set \(X\) is a family \(\mathcal T\subseteq\mathcal P(X)\) such that \(\varnothing,X\in\mathcal T\), arbitrary unions of members of \(\mathcal T\) lie in \(\mathcal T\), and finite intersections of members of \(\mathcal T\) lie in \(\mathcal T\). The pair \((X,\mathcal T)\) is a topological space; members of \(\mathcal T\) are open sets.
Example. For any set \(X\), the discrete topology \(\mathcal P(X)\) remembers every subset as open, while the indiscrete topology \(\{\varnothing,X\}\) remembers only the trivial open sets. On an infinite set, the cofinite topology consists of \(\varnothing\) and all subsets whose complements are finite.
Proposition. The intersection of any collection of topologies on \(X\) is again a topology on \(X\).
Proof. The sets \(\varnothing\) and \(X\) belong to every topology in the collection, hence to their intersection. If a family of sets belongs to the intersection, then it belongs to each topology in the collection; each topology therefore contains its union. The same argument applies to finite intersections.
Closed sets and closure
Closed sets are not extra structure; they are the dual language obtained by complements.
Definition. A subset \(F\subseteq X\) is closed if \(X\setminus F\) is open. For \(A\subseteq X\), the closure \(\overline A\) is the intersection of all closed sets containing \(A\), and the interior \(A^\circ\) is the union of all open sets contained in \(A\).
Proposition. A point \(x\in X\) belongs to \(\overline A\) if and only if every open neighborhood \(U\) of \(x\) meets \(A\).
Proof. If some open neighborhood \(U\ni x\) is disjoint from \(A\), then \(X\setminus U\) is closed, contains \(A\), and does not contain \(x\); hence \(x\notin\overline A\). Conversely, if \(x\notin\overline A\), then \(X\setminus\overline A\) is an open neighborhood of \(x\) disjoint from \(A\).
Corollary. A subset \(A\) is closed exactly when \(A=\overline A\), and it is dense exactly when every nonempty open set meets \(A\).
Proof. The first assertion follows from the definition of \(\overline A\) as the smallest closed set containing \(A\). The second is the preceding neighborhood criterion applied to \(\overline A=X\).
Neighborhoods and local reasoning
Most topological arguments are local: one proves facts by shrinking open neighborhoods.
Definition. A neighborhood of \(x\) is a set \(N\subseteq X\) containing an open set \(U\) with \(x\in U\subseteq N\). A set is open if and only if it is a neighborhood of each of its points.
Proposition. The neighborhood system \(\mathcal N(x)\) satisfies: every \(N\in\mathcal N(x)\) contains \(x\); supersets of neighborhoods are neighborhoods; finite intersections of neighborhoods are neighborhoods; and every \(N\in\mathcal N(x)\) contains a neighborhood \(M\) of \(x\) such that \(N\in\mathcal N(y)\) for all \(y\in M\).
Proof. The first three properties follow directly from the open-set axioms. For the last, choose an open \(M\) with \(x\in M\subseteq N\). If \(y\in M\), then the same open set \(M\) shows that \(N\) is a neighborhood of \(y\).
Comparing topologies
A finer topology has more open sets, so it distinguishes points and limits more sharply.
Definition. If \(\mathcal T_1\subseteq\mathcal T_2\), then \(\mathcal T_2\) is finer than \(\mathcal T_1\), and \(\mathcal T_1\) is coarser than \(\mathcal T_2\).
Proposition. The identity map \(\operatorname{id}: (X,\mathcal T_2)\to (X,\mathcal T_1)\) is continuous if and only if \(\mathcal T_1\subseteq\mathcal T_2\).
Proof. Continuity of the identity means that every open set in the target topology \(\mathcal T_1\) has open inverse image in the source topology \(\mathcal T_2\). The inverse image of any subset under the identity is itself, so this condition is precisely \(\mathcal T_1\subseteq\mathcal T_2\).
Remark. Finer topologies make continuity from the space easier and continuity into the space harder. This simple variance explains many choices of initial and final topologies later.