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A homology orientation chooses an orientation of the real vector space ${⨁}_i{H}_i(M;\mathbb{R})$. It fixes the sign in refined Turaev torsion.

A second-countable topological space has a countable collection of open sets such that every open set is a union of members of that collection. A space covered by countably many open manifold charts is second countable because each coordinate domain has a countable basis of rational balls.

A basis of a topology is a family of open sets such that every open set is a union of members of that family. Equivalently, every point of every open set lies in a basis member contained in it. The rectangles $U\times V$ form a basis for the product topology of two spaces. This topological meaning differs from a linear-algebra basis.

The boundary of a subset $A$ of a topological space $X$ consists of the points whose every neighbourhood in $X$ meets both $A$ and its complement $X\setminus A$. Equivalently,Here the two operations are closure and interior. The second formula shows that the boundary is a closed set and that $A$ and its complement have the same boundary. It locates precisely the points where membership cannot be decided from a small neighbourhood alone. ^[ref]

For $A=[0,1)$ in the usual real line, the interior is $(0,1)$, the closure is $[0,1]$, and the boundary is $\{0,1\}$. In particular, $0$ belongs both to $A$ and to its boundary, while $1$ belongs to the boundary but not to $A$. A boundary point need not be excluded from the original set: open sets exclude all their boundary points, closed sets contain all theirs, and this half-open interval does one of each.

The upper panel visualizes the relative example from open set: for $X=[0,1]$ and $A=[0,1/2)$, an open ball about zero in $X$ is $B_X(0,r)=X\cap (-r,r)=[0,r)$ when $0<r<1/2$. This open ball stays in $A$; points outside $X$ are not required. The middle panel shows that $X$ is clopen in itself, with interior and closure both $X$ and empty boundary, although its boundary in the real line is $\{0,1\}$. The lower panel returns to ambient space $\mathbb{R}$ and separates the interior, closure, and boundary of $[0,1)$; both marked boundary points are included in the boundary itself.

A topological space has local connectedness when every point has a neighborhood basis of connected open sets. In particular, a planar domain has local connectedness, but its domain boundary need not be. Local connectedness of the boundary is the condition governing a continuous extension of a conformal bijection to the parametrizing circle.

Attaching an index-$k$ handle to an $n$-dimensional manifold with boundary glues $D^k\times D^{n-k}$ along $\partial D^k\times D^{n-k}$. The complementary $D^k\times \partial D^{n-k}$ replaces the attaching region in the boundary.

A topological manifold has dimension $n$ when its manifold charts have coordinate domains open in ${\mathbb{R}}^n$. For a smooth manifold, each tangent space has dimension $n$, and a tangent bundle has vector bundle rank $n$.

A smooth manifold carrying a nondegenerate bilinear form on each tangent space, with one negative and all remaining positive eigenvalues, or the overall reversed-sign convention. Its nonzero null vectors have vanishing metric quadratic form.

For a pair $(X,A)$ of CW complexes, relative topological K-theory and Bott periodicity give the cyclic exact sequence

For a nonempty subcomplex $A\subset X$, relative Topological K-theory is $K^i(X,A)={\tilde{K}}^i(X/A)$, where the collapsed subcomplex is the basepoint. It fits into the K-theory six-term exact sequence.

The rank of a vector bundle is locally constant on the base. Taking the difference of the ranks extends to the Grothendieck group of vector bundles, giving the rank map in Topological K-theory. For a finite CW complex, its kernel consists of classes of rank zero on every connected component.

For a finite covering map $p:Y\to X$, define the fiber of the transferred vector bundle by $(p_{!}V{)}_x={⨁}_{y\in p^{-1}(x)}{V}_y$. Local trivializations of the covering map make this a vector bundle, and compatibility with direct sums extends it to Topological K-theory.

If $L$ is the complexification of a real vector bundle of the tautological bundle and $\mu =[L]-1$, thenRealizing Real projective space as the circle bundle of the square of the tautological bundle on Complex projective space gives this through the K-theory Gysin sequence of a sphere bundle.

For a turbulent plume in a homogeneous ambient of density ${\rho }_0$, define actual sectional mass flux, momentum flux and density-weighted buoyancy flux by $Q=\pi b^2\rho w$, $M=\pi b^2\rho w^2$ and $F=\pi b^{2}wg({\rho }_0-\rho )$. With non-Boussinesq Batchelor entrainment $v_e=\alpha w\sqrt{\rho /{\rho }_0}$, their balances areHere $Q^2/M$ is sectional mass, $QF/M$ is sectional density-deficit buoyancy, and the entrained ambient initially has zero vertical momentum. The last equation follows by subtracting mass conservation from ${\rho }_0$ times volume conservation. Scase, Caulfield, Dalziel and Hunt's time-dependent plume model derives the equivalent system with the common factor $\pi$ suppressed.

Dividing the mass flux, momentum flux, and density-weighted buoyancy flux by ambient mass density gives the kinematic fluxes $q=\pi b^{2}V$, $m=\pi b^2{V}^2$, and $f=\pi b^{2}Vg({\rho }_0-\rho )/{\rho }_0$. This convention makes dimensional similarity solutions independent of the arbitrary ambient density unit.

The spinor matrix $B$ obeys $B^{-1}{\gamma }^{\mu \ast }B=t_{\mu }{\gamma }^{\mu }$, with $t_0=1$ and $t_i=-1$. Conjugating the explicit $i$ and three spatial matrices also gives $B^{-1}{\gamma }^{5\ast }B={\gamma }^5$. With spin phases $f_s=(-1{)}^{1/2-s}$ and identities $f_s{u}^{-s\ast }(p_T)={\gamma }^{5}C{u}^s(p)$, relabeling spin and momentum in the mode expansion of a Dirac field gives $B=-{\gamma }^{5}C$, because $f_{-s}=-f_s$. That last relation is phase-convention dependent; the quantum time-reversal operator itself is not just a spinor matrix.

The vector Dirac current has an even temporal component and odd spatial components under time-reversal symmetry. Invariance of its real minimal coupling therefore gives the displayed transformation of the gauge field, up to a gauge transformation. Differentiation at $x_T=(-x^0,\mathbf{x})$ yields $F_{\mu \nu }(x)\mapsto -t_{\mu }{t}_{\nu }{F}_{\mu \nu }(x_T)$, with $t=(1,-1,-1,-1)$. Thus the electric field is even and the magnetic field is odd.

At fixed cylindrical radius $R$ with $|z|\ll R$, the central gravitational potential is harmonic in height. The vertical epicyclic frequency equals ${\Omega }_K$ in this spherical potential.

Kinetic decoupling occurs when momentum-exchange collisions cease to maintain the temperature and velocity dispersion of a particle species in equilibrium with a surrounding bath. Afterwards, a freely moving nonrelativistic particle has momentum proportional to $a^{-1}$ and effective kinetic temperature proportional to $a^{-2}$, where $a$ is the scale factor.

Double-diffusive convection occurs when two properties, such as temperature and composition, affect buoyancy but diffuse at different rates. A stabilizing composition gradient can oppose a destabilizing thermal gradient and change the form of heat transport.