A map between frames preserving every join and every finite meet, including the empty join $0$ and empty meet $1$. Such a map is automatically order-preserving. Inverse image under a continuous map is an example.

A point is a frame homomorphism to the two-element frame $2=\{0<1\}$. Its inverse image of $1$ is a completely prime filter: membership of an arbitrary join forces membership of one term. The point space $P(A)$ has opens $U_a=\{p:p(a)=1\}$. Indeed $U_{a\wedge b}=U_a\cap U_b$ and $U_{{\bigvee }_i{a}_i}={\bigcup }_i{U}_{a_i}$, so these sets already form a topology.

The open subsets of a topological space form a frame. Arbitrary joins are unions, finite meets are intersections, and an arbitrary meet is the interior of an intersection. A continuous map induces a frame homomorphism in the opposite direction by inverse image.

There is a natural bijection between continuous maps $X\to P(A)$ and frame homomorphisms $A\to O(X)$. It sends $f$ to $a\mapsto f^{-1}(U_a)$; the inverse sends $x$ to the point $a\mapsto 1_{\{x\in h(a)\}}$. Its adjunction unit sends a point of $X$ to its open-neighbourhood membership map. The counit is represented by $a\mapsto U_a$. Every open of $P(A)$ is such a $U_a$, and every point of a frame $O(P(A))$ is uniquely evaluation at a point of $A$. Consequently $P(A)\to P(O(P(A)))$ is a homeomorphism, and this is an idempotent adjunction.

A locale is an object of the opposite of the category of frames. A map $f:X\to Y$ is specified by a frame homomorphism $f^{\ast }:\mathcal{O}(Y)\to \mathcal{O}(X)$. This makes open-set algebra primary and allows spaces without enough ordinary points. A point of a frame is the inverse-image map of a point of a locale.

A nucleus is an inflationary, idempotent, finite-meet-preserving map on a frame. Its fixed elements form a frame, whose joins are obtained by applying $j$ to ambient joins. The induced locale is a sublocale. The nuclei on a frame themselves form a frame in pointwise order; their meets are pointwise, but their joins usually are not.

A preframe is a partially ordered set with finite meets and nonempty directed joins, with finite meets distributing over directed joins. Its homomorphisms preserve these operations. Forgetting the arbitrary joins of a frame leaves a preframe.

An upward-closed subset of a frame containing $1$, excluding $0$ and closed under finite meets is completely prime if membership of an arbitrary join implies membership of some summand. Its characteristic map is a point of a frame. Constructively the map takes values in the frame of truth values.

A subframe is a subset of a frame closed under arbitrary ambient joins and finite ambient meets, including $0$ and $1$. The inclusion is a frame homomorphism. In contrast, the fixed elements of a nucleus on a frame usually need their own regularized joins and are not a subframe.

A clique in a simple graph $G$ is a set $C$ of vertices such that every pair of distinct members is adjacent. Equivalently, its induced subgraph $G[C]$ is a complete graph. The surrounding graph need not itself be complete: a clique describes complete pairwise connectivity within a selected group.

Consider the original graph withThe set $\{a,b,c\}$ is a clique because all three pairs $ab,ac,bc$ are edges. The set $\{b,c,d\}$ is not a clique: the edge $bd$ is missing. A path in a graph connecting all selected vertices is therefore insufficient; every pair must have a direct edge.

Here the clique number is $3$. The maximum clique is $\{a,b,c\}$, but $\{c,d\}$ and $\{d,e\}$ are also maximal cliques. This distinction is useful when an algorithm stops as soon as its current clique cannot be enlarged: that stopping condition establishes maximality, while finding the largest possible clique is a separate requirement.

A topological space is compact when every cover by members of a fixed subbase has a finite subcover. Equivalently, every family of complements of subbasic opens with the finite-intersection property has nonempty intersection.

A topological space is compact if and only if every ultrafilter on it converges to at least one point, assuming the ultrafilter lemma. No Hausdorff assumption is needed. In a compact space, the closures of the filter members have the finite intersection property, so choose a point in their intersection. Every open neighbourhood of that point belongs to the ultrafilter, since otherwise its closed complement would be a filter member. Conversely, extend the complements of a cover with no finite subcover to an ultrafilter; a limit point lies in a covering open set, contradicting the simultaneous membership of that set and its complement.

If $T$ is compact and $\lambda \ne 0$, then $T-\lambda I$ is injective exactly when it is surjective. Equivalently, every nonzero point of the spectrum of $T$ is an eigenvalue. Compactness turns an approximate eigenvector sequence into a convergent eigenvector sequence.

A nondegenerate star-algebra of compact operators decomposes its Hilbert space into irreducible closed reducing subspaces, each equivalence class occurring finitely many times. Pass to its norm closure. A nonzero compact positive element supplies a finite-rank spectral projection; a minimal projection in its finite-dimensional corner has scalar corner algebra. A cyclic vector in its range generates an irreducible reducing subspace. A maximal orthogonal family exhausts the space by nondegeneracy. Infinitely many equivalent copies would give orthogonal unit vectors whose images under some compact algebra element have equal nonzero norms, contradicting compactness.

On a bounded open set, let $Tv\in W_0^{1,2}$ be the weak solution of $\Delta Tv=v$. The energy estimate bounds $T:L^2\to W_0^{1,2}$. For $p>n$, put $q=2p/(p-2)<2n/(n-2)$. The zero-boundary Rellich-Kondrachov compactness theorem gives compact embedding into $L^q$, and Hölder's inequality gives $\Vert wz{\Vert }_2\le \Vert w{\Vert }_p\Vert z{\Vert }_q$. Hence $Kv=wTv$ is compact on $L^2$, without a regularity assumption on the boundary. Writing $v=\Delta u$ makes $\Delta u-wu=f$ equivalent to $(I-K)v=f$. The Fredholm alternative for a compact operator then identifies unique solvability with triviality of the homogeneous kernel.

Neglecting curvature and dark energy, integration of the Friedmann equation for radiation in cosmology and pressureless matter gives the displayed comoving particle horizon. At matter-radiation equality, $a_{\mathrm{eq}}={\Omega }_{r,0}/{\Omega }_{m,0}$, this is $2(\sqrt{2}-1)$ times the radiation-only extrapolation $c{a}_{\mathrm{eq}}/(H_0\sqrt{{\Omega }_{r,0}})$. Its associated inverse length differs by an order-one factor from the matter-radiation equality scale $a_{\mathrm{eq}}{H}_{\mathrm{eq}}/c$.

For an Einstein-de Sitter universe normalized by $a(t_0)=1$, $a=(t/t_0{)}^{2/3}$. Its current comoving particle horizon is $3c{t}_0$. A photon emitted inward at that location at $t_0$ covers $3c{t}_0[(t/t_0{)}^{1/3}-1]$ in comoving distance by time $t$. Equating these distances gives arrival at $8t_0$, a travel time of $7t_0$.

A comoving observer has constant comoving coordinates in a homogeneous expanding universe. Its physical separation from another comoving observer grows in proportion to the scale factor, giving relative recession governed by the Hubble parameter. Light exchanged between these observers undergoes cosmological redshift.

For the symmetric group $S_3$, take the transpositions $x=(12)$ and $y=(23)$. Products of permutations apply the rightmost factor first. ThenThe last permutation sends $1\mapsto 3$, $3\mapsto 2$, and $2\mapsto 1$: two swaps fail to commute by an entire three-cycle.

Every group commutator has sign of a permutation $+1$, so $S_3^{\prime }\subseteq A_3$, the alternating group of even permutations. Conversely, the displayed group commutator generates $A_3=\{e,(123),(132)\}$. Therefore $S_3^{\prime }=A_3$. The two cosets are the even permutations and the odd permutations, and their multiplication is just multiplication of signs:This is the first isomorphism theorem applied to the sign of a permutation. Once commutativity is forced, all three transpositions become indistinguishable, and both three-cycles become the identity element. The surviving information is exactly whether the permutation is even or odd. Compare the very different center of a group $Z(S_3)=\{e\}$.

A comet tail is an extended structure formed from material released by a comet. Its dust tail is shaped largely by radiation pressure, while an ion tail is shaped by the solar wind and magnetic field. The cometary dust tail is the solid-grain component.