Suppose a first-order theory characterized equivalence relations with exactly one infinite class. Name two inequivalent points and require each named class to have at least $n$ elements for every positive integer $n$. Every finite subset is satisfied with one infinite class and a sufficiently large finite class. Compactness theorem would produce a first-order model with two infinite classes, a contradiction.

In a Hausdorff space, a point $x$ outside a compact subset $K$ can be separated from all of $K$ by disjoint open sets. Choose disjoint open neighbourhoods $U_k$ of $k$ and $V_k$ of $x$ for each $k\in K$. Finitely many $U_k$ cover $K$ by compactness. Their union and the matching finite intersection of the $V_k$ give the displayed separation. If $K$ is empty, choose $U=\varnothing$, $V=X$. Thus $X\setminus K$ is open, proving that $K$ is closed.

A closed filter base in a compact space has nonempty intersection. If the space is Hausdorff and every base member is connected, the intersection is connected. Otherwise normality separates its two closed components by disjoint open sets. Compactness of the complement supplies finitely many base members whose intersection lies in that union; directedness then gives a single base member meeting both sides, contradicting its connectedness.

The elapsed time between a change at a logic gate's input and the corresponding settled change at its output, under specified loading and transition conditions. Dependent gate delays add along a path. The combinational critical path gives the longest such input-to-output path; reconvergent paths of unequal delay can produce a logic hazard even when the final Boolean function value does not change.

A logic hazard is a transient output error caused by unequal propagation delays along paths in a Boolean circuit. A static hazard temporarily changes an output that should remain constant; a dynamic hazard introduces extra transitions when the output should change once. A steady-state truth table records final values and therefore does not describe these timing effects.

For example, implement $F=ab+\bar{a}c$ with AND gates, an OR gate and a NOT gate. With $b=c=1$, changing $a$ should leave $F=1$. But if the first product falls before the second rises, both OR inputs can momentarily be zero. Adding the redundant term $bc$ gives $F=ab+\bar{a}c+bc$: it does not change the Boolean function, by the consensus theorem, and remains one throughout that single-input transition. This argument assumes the other inputs remain fixed and the circuit begins settled. Simultaneous input changes and more general implementations require their own timing analysis.

A combinational logic circuit mapping a binary input code to selected output lines. In a one-of-$k$ register-address decoder, line $i$ is asserted exactly when the address equals $i$, optionally combined with an enable. This selects which processor register may capture a common write-data word.

Choose $b_n\in (n,n+1/2)\setminus A$, possible because each interval is uncountable. Every tail $K_n=\{b_k:k\ge n\}$ is closed and disjoint from $A$. Therefore $U_n=\mathbb{R}\setminus K_n$ is an open saturated set for collapsing a subset to a point, so its image in the quotient topology is open. These increasing images cover the quotient, but no finite selection covers it, because the largest selected image still misses an infinite tail of distinct singleton classes. This is an open cover with no finite subcover, proving noncompactness. The argument does not require the countable subset to be closed or nondense. For the empty subset, read the quotient as the original line.

For an infinite cardinal number $\kappa$, use cardinal arithmetic with the axiom of choice. If $\lambda =2^{\kappa }$ had cofinality $\mu \le \kappa$, choose cofinal ordinals ${\alpha }_i<\lambda$, $i<\mu$. Their cardinalities $a_i$ satisfy ${\sum }_i{a}_i=\lambda$. The König theorem for cardinal numbers gives $\lambda <{\prod }_i{a}_i^{+}\le {\lambda }^{\mu }\le (2^{\kappa }{)}^{\kappa }=\lambda$, a contradiction.

A cofinal subset $C$ of a partially ordered set $P$ satisfies: every $p\in P$ is at most some $c\in C$. For example, the even natural numbers are cofinal in the natural numbers, because every natural number is bounded above by an even one.

For a nonzero limit ordinal $\alpha$, this condition is equivalent to $\sup C=\alpha$. For a successor ordinal $\beta +1$, it instead requires $\beta \in C$: the singleton $\{\beta \}$ is already cofinal, although its supremum is $\beta$. Thus the equality $\sup C=\alpha$ must not be used without the limit ordinal hypothesis. A convention valid for every ordinal is $\sup \{\gamma +1:\gamma \in C\}=\alpha$.

Consider $a(v,v)={\int }_b^c[|v^{\prime }{|}^2+V{v}^2]$ on real $H^1$ functions with $v(c)=0$, where $V_{-}=\max (-V,0)$. The pointwise estimate in the Endpoint Poincaré inequality gives ${\int }_b^c{V}_{-}{v}^2\le \theta \Vert v^{\prime }{\Vert }_2^2$ with $\theta ={\int }_b^c{V}_{-}(x)(c-x)dx$. If $\theta <1$, then $a(v,v)\ge (1-\theta )\Vert v^{\prime }{\Vert }_2^2\ge (1-\theta )[1+(c-b{)}^2/2{]}^{-1}\Vert v{\Vert }_{H^1}^2$. Thus the form is a coercive bilinear form even though the potential can change sign. For $V(x)=x$ on $(-1,1)$, $\theta ={\int }_{-1}^0(-x)(1-x)dx=5/6$, yielding the explicit constant $1/18$.

A static CMOS implementation of a NAND gate has a pull-down network of NMOS transistors in series and a pull-up network of PMOS transistors in parallel. With $k$ inputs it uses $k$ devices of each kind. All inputs high connect the output to ground; any low input breaks the pull-down path and connects the output to the positive supply through its corresponding p-channel device.

A perfect map is a continuous closed map whose fibers are compact. Surjectivity is sometimes an additional convention; here it is not imposed, and empty fibers are compact. For maps between Tychonoff spaces, the perfect-map characterization by Stone-Čech remainders identifies this property using the unique extension to their compactifications.

For $R>0$ in a normed vector space, define $P_R(0)=0$ and use the displayed formula elsewhere. This continuous map fixes the closed ball of radius $R$ and maps every point outside it to the boundary along its radial line. If a continuous map $T$ is defined only on that closed ball, $T\circ P_R$ is a continuous extension to the whole space. This extension is useful when applying a theorem whose hypotheses concern a globally defined vector field.

The decision clique problem asks whether a given finite graph contains a clique with at least $k$ vertices, where $k$ is part of the input. A proposed clique can be checked by testing all pairs of its vertices for adjacency. The problem is NP-complete; the optimization version asks for a largest clique.

Taking a complement graph turns a clique into an independent set, so clique search and maximum-independent-set search encode the same combinatorial obstruction. For fixed $k$, checking all $k$-vertex subsets takes polynomial time in the graph size, but its exponent depends on $k$. This does not give a polynomial-time algorithm when $k$ varies with the input.

An affine linear classifier assigns binary labels according to the sign of an affine function. Its boundary is an affine hyperplane, and its positive region is a closed half-space under the displayed convention. On ${\mathbb{R}}^d$ the full class has VC dimension $d+1$: arbitrary signs interpolate on $d+1$ affinely independent points, while the intersecting convex hull partition theorem prevents shattering any $d+2$ points. Requiring $b=0$ gives the homogeneous case and a different VC dimension.

Every ideal class of this real quadratic number field contains an integral ideal of ideal norm at most $\sqrt{14}<4$, by the diamond form of the Minkowski convex body theorem. The only possible norms are one, two and three. The sole prime ideal above two has norm two and is principal, generated by $4+\sqrt{14}$, whose field norm is two. The prime three is inert, since $X^2-14$ is irreducible modulo three, so there is no ideal of norm three. Every class therefore has a principal small representative, and the ideal class group is trivial.

This illustrates why the small-ideal bound can prove a class number without listing arbitrary ideals: only prime ideals of a few possible norms need to be tested. Real quadratic fields require the absolute field norm; the generator here happens to have positive norm.

A flow on a directed graph satisfying capacity constraints and a specified net inflow $d_v$ at each vertex. Positive $d_v$ denotes consumption and negative $d_v$ supply. Summing all balances cancels each edge once as inflow and once as outflow, so ${\sum }_v{d}_v=0$ is necessary. With balanced demands and no lower bounds, add a source edge of flow network edge capacity $-d_v$ into every supply vertex and an edge of flow network edge capacity $d_v$ from each demand vertex to a new sink. A maximum flow saturating all auxiliary edges exists exactly when the demands are feasible.

A circular arch of chord length $2a$ described as a smooth single-valued graph has radius $R>a$ and length $2R\arcsin (a/R)$, ranging strictly between $2a$ and $\pi a$. The straight-line limit has length $2a$. At the semicircle limit the endpoint slopes are infinite. Longer circular arcs require overhangs and cannot be graphs in this class.

For real $A,C$ and complex $B$, if $A\ne 0$, completing the square gives $|z+B/A{|}^2=(|B{|}^2-AC)/A^2$. A positive right side gives a circle; zero gives a single point; a negative value gives the empty set. If $A=0$ and $B\ne 0$, the equation is the straight line $2\mathrm{Re}(\overline{B}z)+C=0$. If $A=B=0$, it describes the whole complex plane when $C=0$, and the empty set otherwise. Thus unrestricted coefficients require both the single-point and whole-plane degeneracies.

Every continuous injective map $g:S^1\to S^1$ is a homeomorphism. After normalizing $g(1)=1$ by a rotation, it is homotopic either to the identity or to complex conjugation, and has degree $1$ or $-1$.

To see this without assuming surjectivity, lift $t\mapsto g(e^{2\pi it})$ on $[0,1]$ through the covering map $u\mapsto e^{2\pi iu}$ to a path $G$ with $G(0)=0$. Injectivity makes $G$ injective on $[0,1)$, hence strictly monotone. Its endpoint $d=G(1)$ is an integer. Strict monotonicity excludes $d=0$; if $|d|>1$, the intermediate value theorem gives an interior integer value, contradicting injectivity of $g$. Thus $d=\pm 1$, and $\exp (2\pi i((1-s)G(t)+sdt))$ descends to a homotopy from $g$ to $z\mapsto z^d$. If $g$ omitted a point, a continuous branch of the complex argument on the punctured circle would make it null-homotopic, contradicting this nonzero degree. Hence it is surjective. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism.