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The Klein bottle is obtained from a square by identifying one pair of opposite sides with matching direction and the other pair with reversed direction.

The Möbius band is the topological surface obtained by gluing the ends of a rectangle with a half-turn. Precisely, it is the quotient spaceThe twist identifies the two apparent long edges into one boundary circle. Following the core circle of a Möbius band reverses local handedness, so this surface is not orientable. An untwisted untwisted band has two boundary circles and is orientable: the distinction survives stretching and bending because orientability is a topological invariant.

To explore the construction with paper, mark the two long edges in different colours, join the short ends after a half-turn, and trace an edge. The trace passes from one marked edge to the other without ever leaving the boundary. Separately, trace the centre line and track a local transverse arrow: it comes back reversed after one circuit. The boundary closes after twice the travel around the centre, while the transverse arrow needs two circuits to return to its original direction.

This original mathematical model has central radius $R=2$ and half-width $a=0.55$:Its seam obeys $F(2\pi ,w)=F(0,-w)$. The dotted circle is the core circle of a Möbius band; the teal curve follows the single boundary through $0\le \theta \le 4\pi$. At the same point on the core, the arrows for $\theta =0$ and $\theta =2\pi$ are opposite. The graph compares a continuously followed unit normal with its starting value in the ambient space; the endpoint reversal, rather than any intermediate zero of the graph, establishes the obstruction. It is not a statement about parallel transport.

The Möbius band has the homotopy type of a circle: a deformation retraction shrinks each width coordinate to zero. Nevertheless its boundary represents twice a generator of its fundamental group. The boundary of a Möbius band winds twice around its core, so with compatible generator choices the boundary generator maps to $c^2$ in ${\pi }_1(M)\cong \mathbb{Z}$.

The historical account by Julyan H. E. Cartwright and Diego L. González places the mathematical discovery by August Möbius and Johann Listing in 1858, while investigating representations of twisted bands predating that work. A familiar-looking picture and a recognition of the underlying topological invariant are different kinds of evidence. ^[ref]

The torus is the product of two circles and can be formed by identifying opposite sides of a square.

A polygonal schema constructs a surface by identifying paired polygon edges.

Every simple closed curve on the sphere separates it into two connected components, each having the curve as its boundary.

A free action of a finite group on a topological surface has a surface quotient: sufficiently small coordinate discs have disjoint translates and descend homeomorphically to quotient neighbourhoods.

The Kitaev chain is a one-dimensional spinless p-wave superconductor. Its topological phase has a Majorana zero mode at each end of an open chain.

A quadratic fermion Hamiltonian is bilinear in creation and annihilation operators. In a Majorana basis it is determined, up to a constant, by a real antisymmetric matrix.

An open cover of a subset $A$ is a family of open sets whose union contains $A$.

The Sierpiński space is $\{0,1\}$ with open sets $\varnothing$, $\{1\}$, and $\{0,1\}$. Continuous maps $X\to S$ are in bijection with open subsets of $X$ through $g\mapsto g^{-1}(1)$.

A family $\mathcal{T}$ of subsets of a set $X$ is a topology when $\varnothing ,X\in \mathcal{T}$, every union of members of $\mathcal{T}$ belongs to $\mathcal{T}$, and every finite intersection of members of $\mathcal{T}$ belongs to $\mathcal{T}$.

The circle is the subspace $S^1=\{z\in \mathbb{C}:|z|=1\}$, equivalently the quotient of a closed interval obtained by identifying its endpoints.

The unitary group $U(n)$ consists of complex matrices preserving the standard Hermitian inner product.

For loops based at the identity of a topological group, pointwise multiplication and concatenation obey the interchange law and have the same identity. The Eckmann-Hilton argument therefore makes the fundamental group abelian.

For a two-component gapped BdG Hamiltonian constrained to a plane, the winding number counts how many times its coefficient vector encircles the origin as crystal momentum traverses the Brillouin zone.

When pressure and velocity are in quadrature, their instantaneous energy flux oscillates but its period average vanishes.

For $N$ identical sites of spacing $a$ with periodic boundary conditions, translation eigenstates have coefficientsThus $k=2\pi j/(Na)$ modulo the reciprocal-lattice period $2\pi /a$.

The Hills mass is the black-hole mass above which a given star crosses the capture boundary before being tidally disrupted. For a Schwarzschild capture radius $2G{M}_{\mathrm{BH}}/c^2$, it is $c^3{R}_{\ast }^{3/2}/(2G^{3/2}{M}_{\ast }^{1/2})$ with the corresponding tidal-radius convention.

For a star of mass $M_{\ast }$ and radius $R_{\ast }$ near a black hole of mass $M_{\mathrm{BH}}$, the Newtonian tidal disruption radius is $r_t\simeq R_{\ast }(2M_{\mathrm{BH}}/M_{\ast }{)}^{1/3}$ under a peak-to-center tidal-acceleration convention.

The surface density of a disk is its volume density integrated vertically, $\Sigma ={\int }_{-\infty }^{\infty }\rho dz$.