Every function $f:\mathbb{R}\to [\mathbb{R}{]}^{<{\omega }_1}$ has reals $x,y$ with $x\notin f(y)$ and $y\notin f(x)$. The values are countable subsets. Requiring distinct witnesses is equivalent, by first adjoining $x$ to $f(x)$. The Freiling theorem identifies this axiom with the negation of the Continuum hypothesis in ZFC.

Place one distinct label at each vertex of a finite connected weighted graph. At rate $c_{uv}=c_{vu}\ge 0$, interchange the labels at $u$ and $v$. This continuous-time Markov chain has a uniform distribution on a finite set as its stationary distribution. The position of one label is the random walk with the same rates.

A jump process with event types $k$, increments ${\nu }_k$, and rates $a_k(X)$ can be represented as $X(t)=X(0)+{\sum }_k{\nu }_k{P}_k({\int }_0^t{a}_k(X(s-))ds)$, with independent unit-rate Poisson processes. This represents the Markov jump-process generator through time-changed counts.

Global balance equates total stationary incoming and outgoing transition intensities at each state. Detailed balance for a continuous-time Markov chain is sufficient but stronger: it equates each pair separately.

A continuous-time Markov chain is nonexplosive when it makes only finitely many jumps in every finite time interval, almost surely. Uniformly bounded total jump rates suffice. Another useful criterion bounds the states visited over each finite interval by a random finite set on which rates are bounded.

For a finite irreducible Markov chain in continuous time, time averages converge almost surely to averages under its unique stationary distribution, from any initial state. Applying this to an activation indicator yields the corresponding throughput.

For a finite-state continuous-time Markov chain, a transition intensity matrix has nonnegative off-diagonal rates $q_{rs}$ and $q_{rr}=-{\sum }_{s\ne r}{q}_{rs}$. Every row sums to zero. In the time-homogeneous case its transition probability matrix is the matrix exponential $P(t)=e^{tQ}$. An absorbing state has a zero row.

For a normal operator with disconnected spectrum $K=K_1\bigsqcup K_2$, the indicator function of a nonempty proper clopen component union is continuous on $K$. The continuous functional calculus makes it a nonzero proper self-adjoint linear projection commuting with $T,T^{\ast }$. Its range is therefore a nontrivial closed reducing subspace. No eigenvector is needed.

A function is lower semicontinuous when its strict superlevel sets are open, equivalently when its sublevel sets are closed. In Euclidean spaces this is the sequential lower semicontinuity condition $f(x)\le {\mathrm{liminf}}_{k}f(x_k)$ whenever $x_k\to x$. This closedness property is essential to existence of proximal minimizers.

If $|h(x)|\le C$ near $c$ and $a(x)\to 0$ as $x\to c$, then $a(x)h(x)\to 0$ by the squeeze theorem, even if $h$ has no limit. Defining the product to be zero at $c$ gives continuity there. For example, $\sqrt{|x|}\sin (1/\sin x)$ extends continuously at zero; the same oscillation near a nonzero multiple of $\pi$ has nonvanishing amplitude and does not have a continuous zero extension.

A real or extended-real function is upper semicontinuous when every strict sublevel set is open. Equivalently its negative is lower semicontinuous. An arbitrary infimum of continuous functions is upper semicontinuous, even when the infimum is over an uncountable family; its strict sublevel sets are unions of the corresponding open sublevel sets.

For a uniform prestate in $N+1$ dimensions and poststate proportional to ${\sum }_{i=1}^N|i⟩-(N-1)|N+1⟩$, each binary question $P_i$ versus $I-P_i$ has conditional probability one for $i\le N$. A fully resolved quantum measurement instead assigns $1/D$ to each of the first $N$ outcomes and $(N-1{)}^2/D$ to the last. The context dependence of pre- and post-selected measurements explains why these alternative certainties do not imply simultaneous properties.

The constructible universe satisfies the diamond principle. Together with antichain sealing by diamond, this supplies a Suslin tree in $L$.

ZFC proves that the constructible universe is an inner model of ZFC and satisfies the Generalized continuum hypothesis. Relativizing to $L$ yields the corresponding relative-consistency implication.

To separate a subset of a constructible set by a formula interpreted in the constructible universe, apply the reflection theorem for definable hierarchies to the formula and its subformulas. At a sufficiently large reflecting level containing the parameters, the desired subset is definable and belongs to the next constructible hierarchy level. The reflection proof uses least witness stages and ambient Axiom schema of replacement, so does not presuppose Axiom schema of separation within the constructible universe.

The downstream edge of a continuously fed porous gravity current with background flow has a transition width proportional to $\sqrt{a{h}_{\mathrm{int}}t}$ around drift position $ut$. Its similarity solution satisfies $(FF\prime )\prime +\eta F\prime /2=0$, $F(-\infty )=1$, and $F({\eta }_N)=0$. A finite front has $F\sim ({\eta }_N/2)({\eta }_N-\eta )$. A clipped linear ramp is a useful mass-preserving approximation, but not an exact solution of this nonlinear transition equation.

A vertical cut between consecutive columns of $T_k$ contains $2k+1$ edges. All are closed with probability $(1-p{)}^{2k+1}$. Cuts in different columns use disjoint edges, so their closed-cut events are independent. A connection from column zero to column $n$ must cross each of the intervening $n$ cuts. ConsequentlyThis finite-width obstruction persists even when the unrestricted planar percolation is supercritical.

Let $q(n)$ be the unrestricted planar two-point percolation two-point connection probability. Every finite connecting graph path is contained in some strip, so $q_k(n)\uparrow q(n)$ for each fixed $n$. ThereforeThe equality uses commuting infima and monotonicity, not an unjustified interchange of two general limits. The right-hand side is the whole-plane connection decay rate by the Fekete lemma.

For the standard complex orientations, the two summand classes have zero product and equal squares, the oriented degree-four class. The quadratic relations imply all cubic monomials vanish. The rational Betti numbers are $1,2,1$ in degrees zero, two and four.

For closed connected oriented $d$-manifolds with $d\ge 2$,Deleting an open ball changes the Euler characteristic by $-1+\chi (S^{d-1})$, and gluing subtracts $\chi (S^{d-1})$. Since $\chi (S^{d-1})+\chi (S^d)=2$, the formula follows. Thus a connected sum of oriented manifolds subtracts two from the sum of Euler characteristics in even dimension and subtracts zero in odd dimension.