If an orientable three-manifold has nonzero ${\pi }_2$, it contains an embedded essential two-sphere. Consequently a closed orientable irreducible three-manifold with infinite fundamental group is aspherical: its simply connected noncompact universal cover has zero second homotopy and top ordinary homology, hence is acyclic and contractible by the Hurewicz theorem and Whitehead theorem.

If a boundary surface inclusion has noninjective fundamental group map, there is a compression disk whose boundary represents a nontrivial element of that kernel. More generally the boundary can be required to avoid a prescribed normal subgroup if a singular disk with that property exists.

A closed simply connected three-manifold is homeomorphic to the three-sphere. Now a theorem, this identifies every homotopy three-sphere with the standard sphere and removes homotopy-sphere ambiguities in three-dimensional connected-sum classification. Smooth structures are also unique in dimension three.

A three-ball is a three-manifold homeomorphic to the closed unit ball in ${\mathbb{R}}^3$. Its boundary is a two-sphere. Attaching a three-ball to a spherical boundary component is called capping that boundary; marked endpoints may be joined by a straight arc inside the cap.

A punctured three-sphere is $S^3$ with the interiors of finitely many disjoint piecewise linear balls removed. Capping its spherical boundary components with balls restores $S^3$. Removing another ball or gluing two such manifolds along one spherical boundary component preserves this class.

Relative to a triangulation, a normal surface meets every tetrahedron in disjoint normal disks: four types of triangles cutting off vertices and three types of quadrilaterals separating opposite pairs of edges. A normal disk meets any edge at most once; its face arcs join distinct edges. Two different quadrilateral types cannot coexist in a tetrahedron.

Assume the axiom of choice and suppose a poset satisfying the hypothesis has no maximal element. Fix a choice function on its nonempty subsets. By transfinite recursion, choose at each ordinal stage an upper bound of the chain in a partial order already constructed, then choose a strictly larger member. Both choices are available, including the initial empty stage. This produces distinct members throughout the Hartogs ordinal of $P$, contradicting the Hartogs theorem. Thus a maximal element exists. Conversely, apply Zorn's lemma to partial choice functions on a family of nonempty sets, ordered by extension. A chain in a partial order has the compatible union as an upper bound, and a maximal element must be defined on the whole family, since one missing value could be adjoined.

Zorn's lemma was introduced in 1935 to shorten algebra proofs using choice or the well-ordering theorem.^[ref]

For an explicit basis extension in ${\mathbb{R}}^2$, start with $A=\{u\}$, where $u=(1,1)$. Its linear span is the line $y=x$. Adding $w=(1,-1)$ preserves linear independence, because $w$ is outside that line, and gives a basis. Every vector has the finite linear combinationThe vector $2u=(2,2)$ cannot be added while preserving linear independence: it is already in the linear span. Many choices outside the line work, so maximality does not select a unique basis.

For a finite-dimensional vector space with a supplied finite spanning set, a greedy algorithm scans the list, retaining each vector precisely when it is outside the linear span of those already retained. Starting with a linearly independent $A$, the same scan adjoins only the needed vectors. Retained vectors remain linearly independent, and every skipped vector is in the final linear span, so the result is a basis containing $A$. Finite reasoning needs no axiom of choice.

For an arbitrary vector space, this finite stopping argument is unavailable. Accepting Zorn's lemma asserts existence of a maximal element once every chain in a partial order has an upper bound; it supplies no general computational recipe. Over ZF, Zorn's lemma, the axiom of choice and the well-ordering theorem are equivalent. Particular infinite vector spaces can still have explicit bases, such as $1,x,x^2,\dots$ for the vector space of univariate polynomials over a specified field; infinity alone does not force a nonconstructive argument.^[ref]

Every vector space $V$ over a field $F$ has a basis, assuming the axiom of choice. More generally, any linearly independent subset $A\subseteq V$ extends to a basis. This is an algebraic Hamel basis: each representation uses a finite linear combination, even when the basis is infinite.^[ref]

Use the poset of linearly independent subsets containing $A$:It is a set, being a subset of the power set of $V$, and is nonempty because $A\in {\mathcal{P}}_A$.

A short proof outline supplies both essential steps:

Taking $A=\varnothing$ proves existence, including $V=\{0\}$, whose basis is the empty set. The chain in a partial order need not be indexed by the natural numbers: only the finitely many members needed for a particular linear relation are compared. An arbitrary union of linearly independent sets can fail: $\{v\}$ and $\{2v\}$ for a nonzero vector over the real numbers are each linearly independent, but their union is linearly dependent. The nesting of a chain in a partial order is what prevents this failure.

Consider the following finite set of subsets, ordered by subset containment:Every nonempty chain in a partial order has a greatest element within that chain in a partial order, hence an upper bound in $P$. For the empty chain in a partial order, choose $\varnothing \in P$. Thus the hypothesis of Zorn's lemma holds. The two maximal elements are $\{a,b\}$ and $\{b,c\}$: neither is a proper subset of another member of $P$. There is no greatest element, because it would have to contain both, and no member of $P$ contains $\{a,b,c\}$.

For the chain in a partial order $C=\{\varnothing ,\{b\}\}$, the member $\{b\}$ is already an upper bound, although it is not a maximal element of $P$. Both orange members are also upper bounds for $C$. A chain in a partial order need not have a unique upper bound, and the upper bound used to check the hypothesis need not be the eventual maximal element.

The restriction to chains in a partial order matters. The subset $\{\{a\},\{c\}\}\subseteq P$ has no upper bound in $P$, but its two members are incomparable, so it is not a chain in a partial order. Requiring every subset of $P$ to have an upper bound would be much stronger: taking that subset to be $P$ itself would force a greatest element.

For a finite matrix zero-sum game, the value is the payoff that the maximizing player can guarantee and the minimizing player can prevent the payoff from exceeding, using mixed strategies. The common value of the two displayed optimizations exists by the minimax theorem. A pair of strategies can be verified directly by checking that every pure column pays the row mixture at least $v$ and every pure row earns at most $v$ against the column mixture.

With choices ordered rock, scissors, paper and winner payoff $1$, the row-player payoff matrix is $\left(\begin{array}{ccc}0& 1& -1\\ -1& 0& 1\\ 1& -1& 0\end{array}\right)$. Both the row and column sums vanish. Thus the uniform mixed strategy guarantees expected payoff zero against every opposing strategy; the value of a zero-sum game is zero. A player responding to a known nonuniform mixture can instead optimize its linear expected payoff over pure responses and their ties.

The zero set of a function $f$ is the preimage $f^{-1}(\{0\})$, consisting of all its zeros of a function. For a real- or complex-valued continuous function, it is a closed set, because the preimage of the closed singleton $\{0\}$ under a continuous function is closed. A nonzero holomorphic function on a connected open set has isolated zeros by the identity theorem for holomorphic functions.

A definable class bijection $\pi$ of a model's universe changes membership as displayed. Images and inverse images of sets under $\pi$ remain sets by the Axiom schema of replacement. The new extension of $y$ is the old set $\pi (y)$. Thus the representative of any desired extension $b$ is ${\pi }^{-1}(b)$. Extensionality follows from bijectivity; separation and replacement translate to old definable formulas; pairs, unions and power sets have explicit representatives. Infinity follows by iterating the new empty set and successor. Foundation need not survive: interchanging $\varnothing$ and $\{\varnothing \}$ makes the old empty set a Quine atom. A full such permutation of a model of choice still satisfies choice; failure of choice requires an additional symmetry restriction.

Full intuitionistic Zermelo–Fraenkel set theory uses intuitionistic first-order logic, extensionality, pairing, union, infinity, full separation, power set, collection and epsilon induction. Collection is stronger than replacement intuitionistically. This is distinct from predicative constructive ZF, which restricts separation and replaces full power set. The full theory supports a Boolean-valued inner universe over intuitionistic set theory, giving the relative consistency of classical ZF.

The usual set theory with extensionality, empty set, pairing, union, power set, infinity and full separation, before adjoining replacement. Modern presentations often include foundation; the convention should be specified when comparing theories. Class-theoretic variants add classes and class comprehension, rather than treating proper classes as sets.

Zenith distance is the angle from the zenith to a viewing direction: $z=\arccos (\mathbf{n}\cdot {\mathbf{e}}_U)=90^{\circ }-\gamma$, where $\gamma$ is altitude and $0\le z\le 180^{\circ }$. The astronomical horizon has $z=90^{\circ }$, while the zenith has $z=0$. This unsigned separation supplies no azimuth and is distinct from the signed north/south angular readout used under declination.

Three coordinate axes in affine three-space have coordinate ideal $(xy,xz,yz)$; their common point has embedding dimension three. Three distinct coplanar lines meeting at one point have embedding dimension two there. An isomorphism preserves the unique point lying on all three irreducible components and preserves the algebraic cotangent space, so these two configurations are not isomorphic despite identical component incidence.

For a smooth algebraic surface defined by a homogeneous polynomial $F$ in projective space of dimension three, its tangent plane at $P$ is the displayed plane. Every projective line on the surface through $P$ lies in this plane.

Saccharomyces cerevisiae is a particular species of yeast, used in baking, brewing and laboratory research. During ordinary vegetative growth it reproduces by budding: a smaller daughter grows from the mother before separation. Laboratory strains can be maintained as either haploid or diploid biological cells. This makes it possible to investigate genes in a single eukaryote, then use controlled crosses and meiosis to connect a genotype to a phenotype. See Burgess, Powers and Mell, Budding Yeast Saccharomyces cerevisiae as a Model Genetic Organism (2017), pp. 1–2.

The practical value of Saccharomyces cerevisiae as a model organism is the combination of easy laboratory cultivation, accessible genetics, and cellular functions shared with other eukaryotes. A haploid strain has one set of chromosomes, so a changed allele is not accompanied by a second copy at that genetic locus. But a mutation need not produce an observable phenotype under every condition. The yeast is also a whole organism, not a miniature human: transferring a result requires evidence that the relevant function is shared in the target system. See Duina, Miller and Keeney, Budding Yeast for Budding Geneticists (2014): cultivation, genetic tools and conserved functions.

For a concrete example, Hartwell, Culotti and Reid's 1970 work followed temperature-sensitive Saccharomyces cerevisiae strains with successive microscope photographs. Failure to complete a particular part of the cell cycle helped identify required gene functions. Their distinction between when a function is required and where a cell finally stops is especially instructive: the visible endpoint need not be the time of the underlying defect. See Hartwell, Culotti and Reid, Genetic control of the cell-division cycle in yeast. I. Detection of mutants (1970): primary abstract. This is a useful way to turn an observable change in a biological cell into a question about genes and causal order.

In alcoholic fermentation, Saccharomyces cerevisiae converts sugars into ethanol and carbon dioxide. The carbon dioxide helps dough rise; the ethanol is useful in alcoholic drinks. The metabolism also supports production of ATP, a usable cellular energy carrier. This is one metabolic ability of one species, not the definition of every yeast. See Clark, Douglas and Choi, Biology 2e, 7.5: alcohol fermentation and the Burgess, Powers and Mell account above.

For glucose, the ideal net stoichiometry isOne molecule of glucose supplies six carbon chemical atoms: four end up in the two ethanol molecules and two in the two carbon dioxide molecules. The twelve hydrogen and six oxygen chemical atoms also balance. This is a net material balance across several enzyme-catalyzed steps, not a single chemical reaction step or the complete bookkeeping for cellular energy carriers. The standard pathway accounting yields a net two ATP per glucose: the initial breakdown produces the ATP, while the later conversion to ethanol regenerates a coenzyme needed to keep the breakdown going. See Clark, Douglas and Choi, Biology 2e, 7.2: net ATP accounting and the section 7.5 cited above.

At complete conversion, this toy gives $20\mathrm{mmol}$ of ethanol and $20\mathrm{mmol}$ of carbon dioxide. The carbon balance is $6\times 10=2\times 20+1\times 20=60\mathrm{mmol}$ of carbon chemical atoms. A real culture also builds cellular material, can use other metabolic routes, and may retain dissolved carbon dioxide; the toy is a reference balance rather than a prediction of collected gas or beverage yield.

Although alcoholic fermentation does not require molecular oxygen as a reactant, the presence of molecular oxygen does not force Saccharomyces cerevisiae to stop fermenting. De Deken's 1966 experiments reported predominantly fermentative glucose use during exponential growth in air. Consequently, bubbles do not establish either absence of molecular oxygen or an exclusively fermentative metabolism. See De Deken, The Crabtree Effect: A Regulatory System in Yeast (1966): primary abstract.