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A surjective linear isometry $U:C(K)\to C(L)$ between compact Hausdorff function spaces has the form $(Uf)(s)=u(s)f(h(s))$, where $h:L\to K$ is a homeomorphism and $u$ is continuous with modulus one. Thus the Banach space structure of $C(K)$ determines $K$ up to homeomorphism.

The radial map $z\mapsto z/(1+|z|)$ is a homeomorphism from $\mathbb{C}$ to the unit disc, with inverse $w\mapsto w/(1-|w|)$. A conformal equivalence in that direction would be a nonconstant bounded entire function, contradicting the Liouville theorem.

For a T0 space $X$, evaluation against all continuous maps $g:X\to S$ is a homeomorphism onto its image under the mapinto a power of the Sierpiński space. The $T_0$ axiom makes it injective, and the coordinate inverse images of $\{1\}$ recover every open subset of $X$.

A fiber monodromy is the return homeomorphism of the fiber of a bundle over $S^1$. Its mapping torus reconstructs the total space.

If $h:A\to B$ is a homeomorphism and $A$ has the fixed-point property, then a continuous $f:B\to B$ induces $h^{-1}fh:A\to A$. A fixed point of the conjugate map is carried by $h$ to a fixed point of $f$.

On the torus ${\mathbb{R}}^d/{\mathbb{Z}}^d$, an integer matrix $A$ defines an endomorphism. The coordinate circles give a basis of the first homology group, and the induced map on that group is $A$. The Künneth theorem identifies higher integral homology with the exterior powers of ${\mathbb{Z}}^d$, so the degree-$q$ map is ${\bigwedge }^{q}A$. In top degree it is multiplication by $\det A$. The map is a homeomorphism precisely when $A$ is a unimodular matrix; a merely nonsingular integer matrix need not have an integer inverse.

A continuous bijection that is an open map is a homeomorphism. If $g=f^{-1}$ and $U$ is open in the original domain, then $g^{-1}(U)=f(U)$ is open by the open-map property. This proves continuity of the inverse directly.

On a two-sided full shift, the left shift is the homeomorphismIts inverse sends $z(j)$ to $z(j-1)$. On a one-sided sequence space, the same forward shift is generally not invertible.

A local trivialization of a fiber bundle $p:E\to B$ over an open set $U\subseteq B$ is a homeomorphism $p^{-1}(U)\cong U\times F$ commuting with the projections to $U$. For a vector bundle, it must be linear on each fiber. Compatible local trivializations specify the bundle structure and give its transition functions on overlaps.

The mapping torus of $f:F\to F$ isIf $f$ is a homeomorphism, the projection to the circle $S^1$ is a fiber bundle with fiber $F$. For a general continuous map, the mapping torus need not be a fiber bundle, but its homology still has the Wang sequence.

For a continuous self-map $f:X\to X$, the mapping torus has the Wang sequence even when $f$ is not a homeomorphism. Cover the time circle by a seam neighborhood and an interval away from the seam. Their inverse images retract to $X$: the seam neighborhood is a mapping cylinder, and the other is a product. Their intersection retracts to two copies of $X$. The Mayer–Vietoris sequence map on homology is, after consistent choices of signs,Invertible changes of coordinates reduce it to the direct sum of $\mathrm{id}-f_{\ast }$ and an isomorphism. Canceling the isomorphism summands gives the Wang sequence. This proof does not assume a fiber bundle structure for a noninvertible $f$.

Apply an orientation-reversing homeomorphism of $S^3$ to a link and transport its component orientations. Reflection through the projection plane produces a link diagram by switching every crossing.

A nilpotent thickening is a closed immersion defined by a nilpotent ideal sheaf of a closed subscheme. Its underlying topological map is a homeomorphism, because every prime ideal contains every nilpotent element. For example, quotienting the dual numbers by $(\epsilon )$ changes the structure sheaf without changing the single point of the spectrum.

With a consistent surgery orientation convention, $L(p,q)$ and $L(p,q^{\prime })$ are orientation-preservingly related by a homeomorphism exactly when $q^{\prime }\equiv q^{\pm 1}(\mathrm{mod}p)$. An orientation-reversing homeomorphism instead requires $q^{\prime }\equiv -q^{\pm 1}(\mathrm{mod}p)$.