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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, $\Sigma ={\int }_{-\infty }^{\infty }\rho dz$.

The ideal Diesel cycle consists of reversible adiabatic compression, constant-pressure heat addition, reversible adiabatic expansion, and constant-volume heat rejection.

The ideal Otto cycle has adiabatic compression, constant-volume heat addition, adiabatic expansion, and constant-volume heat rejection. For an ideal gas with compression ratio $r=V_1/V_2$ and specific-heat ratio $\gamma$,