Continuity is the structure-preserving notion for topological spaces. A homeomorphism is an isomorphism of topological spaces, so all purely topological properties are invariant under it.
Equivalent forms of continuity
Continuity can be read from open sets, closed sets, closures, or neighborhoods.
Definition. A map \(f:X\to Y\) is continuous if \(f^{-1}(V)\) is open in \(X\) for every open set \(V\subseteq Y\).
Proposition. For \(f:X\to Y\), the following are equivalent: \(f\) is continuous; inverse images of closed sets are closed; \(f(\overline A)\subseteq\overline{f(A)}\) for every \(A\subseteq X\); and for every \(x\in X\), every neighborhood of \(f(x)\) pulls back to a neighborhood of \(x\).
Proof. The closed-set version follows by complementing inverse images. If \(f\) is continuous and \(x\in\overline A\), then every neighborhood of \(f(x)\) has inverse image a neighborhood of \(x\), hence meets \(A\); its image meets \(f(A)\), so \(f(x)\in\overline{f(A)}\). The closure condition applied to \(A=f^{-1}(F)\) with \(F\) closed gives \(f(\overline A)\subseteq F\), hence \(\overline A=A\), so \(A\) is closed. The neighborhood form is just the definition localized at a point.
Composition and categorical behavior
Continuous maps compose, which is why topological spaces form a category.
Proposition. If \(f:X\to Y\) and \(g:Y\to Z\) are continuous, then \(g\circ f:X\to Z\) is continuous. The identity map on every topological space is continuous.
Proof. For an open set \(W\subseteq Z\), \((g\circ f)^{-1}(W)=f^{-1}(g^{-1}(W))\). Since \(g^{-1}(W)\) is open in \(Y\) and \(f\) is continuous, this inverse image is open in \(X\). The identity case is immediate because inverse images under the identity do nothing.
Remark. This observation is modest but decisive: topological constructions are designed by universal properties involving continuous maps.
Homeomorphisms and invariants
Homeomorphic spaces may look different externally, but their open-set structures are identical.
Definition. A homeomorphism \(f:X\to Y\) is a bijection such that both \(f\) and \(f^{-1}\) are continuous. Spaces \(X\) and \(Y\) are homeomorphic if such a map exists.
Proposition. A bijection \(f:X\to Y\) is a homeomorphism if and only if \(U\subseteq X\) is open exactly when \(f(U)\subseteq Y\) is open.
Proof. If \(f\) is a homeomorphism, open sets in \(X\) have open images because \(f(U)=(f^{-1})^{-1}(U)\), and open sets in \(Y\) have open inverse images by continuity of \(f\). Conversely, if the open-set equivalence holds, then inverse images under \(f\) and under \(f^{-1}\) preserve open sets, so both maps are continuous.
Example. The interval \((0,1)\) is homeomorphic to \(\mathbb R\) via \(x\mapsto \tan(\pi(x-\frac12))\). Boundedness and length are not topological properties; openness, connectedness, and compactness behavior are.
Open and closed maps
A continuous bijection need not be a homeomorphism unless one controls images of open or closed sets.
Definition. A map \(f:X\to Y\) is open if it sends open subsets of \(X\) to open subsets of \(Y\), and closed if it sends closed subsets to closed subsets.
Proposition. A continuous bijection that is open or closed is a homeomorphism.
Proof. Let \(f:X\to Y\) be a continuous bijection. To show \(f^{-1}\) is continuous, it is enough to show inverse images of open sets under \(f^{-1}\) are open; these inverse images are exactly sets of the form \(f(U)\) with \(U\) open in \(X\). Thus an open bijection is a homeomorphism. If \(f\) is closed, apply the same argument to closed sets.
Remark. The compact-Hausdorff theorem is a major instance: a continuous bijection from a compact space to a Hausdorff space is closed, hence a homeomorphism.