Kernel (function)

In mathematics,kernel offunction fan equivalence relation onfunction's domain that roughly expressesidea"equivalent as far asfunction f can tell". Note that thereseveral other meanings ofword "kernel"mathematics; see Kernel (mathematics)these.

Forformal definition, let XY be setslet f befunction from XY. If xx'elementsX, then xx'equivalent if f(x)f(x')equal as elementsY. The kernelf isequivalence relation thus defined.

The kernel may be denoted "=_f" (orvariation)may be defined symbolically as

Like any equivalence relation,kernel can be modded out byformquotient set. This quotient set X/=_fcalledcoimage offunction f,denoted "coim f" (orvariation). The coimagenaturally isomorphic (inset-theoretic sense ofbijection) toimage, im f; specifically,equivalence classxX (whichan elementcoim f) correspondsf(x)Y (whichan elementim f).

Like any binary relation,kernel offunction may be thoughtassubset ofCartesian product X × X. In this guise,kernel may be denoted "ker f" (orvariation)may be defined symbolically as

But thisnot useful merely asformalisationset theory! In fact,study ofpropertiesthis subset can shed important light onfunctionquestion. We give here two examples.

First, if XYalgebraic structuressome fixed type (such as groupss, ringss, or vector spaces),iffunction f from XY ishomomorphism, then ker f will besubalgebra ofdirect product X × X. SubalgebrasX × X thatalso equivalence relations (called congruence relations)importantabstract algebra, becausedefinemost general notionquotient algebra. Thuscoimagef isquotient algebraX much asimagef issubalgebraY; andbijection between them becomes an isomorphism inalgebraic sense as well (this ismost general form offirst isomorphism theoremalgebra). The usekernelsthis contextdiscussed further inarticle Kernel (algebra).

Secondly, if XYtopological spacesf iscontinuous function between them, thentopological propertiesker f can shed light onspaces XY. For example, if Y isHausdorff space, then ker f must beclosed set. Conversely, if XHausdorffker f isclosed set, thencoimagef, if givenquotient space topology, must also be Hausdorff.