Quotient mapping

A mapping $f$ of a topological space $X$ onto a topological space $Y$ for which a set $v\subseteq Y$ is open in $Y$ if and only if its pre-image $f^{-1}(v)$ is open in $X$. If one is given a mapping $f$ of a topological space $X$ onto a set $Y$, then there is on $Y$ a strongest topology $\mathcal{T}_f$ (that is, one containing the greatest number of open sets) among all the topologies relative to which $f$ is continuous. The topology $\mathcal{T}_f$ consists of all sets $v\subseteq Y$ such that $f^{-1}(v)$ is open in $X$. This topology is the unique topology on $Y$ such that $f$ is a quotient mapping. Therefore $\mathcal{T}_f$ is called the quotient topology corresponding to the mapping $f$ and the given topology $\mathcal{T}$ on $X$.

The construction described above arises in studying decompositions of topological spaces and leads to an important operation — passing from a given topological space to a new one — a decomposition space. Suppose one is given a decomposition $\gamma$ of a topological space $(X,\mathcal{T})$, that is, a family $\gamma$ of non-empty pairwise-disjoint subsets of $X$ that covers $X$. Then a projection mapping $\pi:X\to\gamma$ is defined by the rule: $\pi(x)=P\in\gamma$ if $x\in P\subseteq X$. The set $\gamma$ is now endowed with the quotient topology $\mathcal{T}_\pi$ corresponding to the topology $\mathcal{T}$ on $X$ and the mapping $\pi$, and $(\gamma,\mathcal{T}_\pi)$ is called a decomposition space of $(X,\mathcal{T})$. Thus, up to a homeomorphism a circle can be represented as a decomposition space of a line segment, a sphere as a decomposition space of a disc, the Möbius band as a decomposition space of a rectangle, the projective plane as a decomposition space of a sphere, etc.

The following properties of quotient mappings, connected with considering diagrams, are important: Let $f:X\to Y$ be a continuous mapping with $f(X)=Y$. Then there are a topological space $Z$, a quotient mapping $g:X\to Z$ and a continuous one-to-one mapping (that is, a contraction) $h:Z\to Y$ such that $f=h\circ g$. For $Z$ one can take the decomposition space $\gamma=\left\{f^{-1}(y):y\in Y\right\}$ of $X$ into the complete pre-images of points under $f$, and the role of $g$ is then played by the projection $\pi$. Suppose one is given a continuous mapping $f_2:X\to Y_2$ and a quotient mapping $f_1:X\to Y_1$, where the following condition is satisfied: If $x',x''\in X$ and $f_1(x')=f_1(x'')$, then also $f_2(x')=f_2(x'')$. Then the unique mapping $g:Y_1\to Y_2$ such that $g\circ f_1=f_2$ turns out to be continuous. The restriction of a quotient mapping to a subspace need not be a quotient mapping — even if this subspace is both open and closed in the original space. The Cartesian product of a quotient mapping and the identity mapping need not be a quotient mapping, nor need the Cartesian square of a quotient mapping be such. The restriction of a quotient mapping to a complete pre-image does not have to be a quotient mapping. More precisely, if $f:X\to Y$ is a quotient mapping and if $Y_1\subseteq Y$, $X_1=f^{-1}(Y_1)$, $Y_1=f|_X$, then $f_1:X_1\to Y_1$ need not be a quotient mapping. This cannot occur if $Y_1$ is open or closed in $Y$.

These facts show that one must treat quotient mappings with care and that from the point of view of category theory the class of quotient mappings is not as harmonious and convenient as that of the continuous mappings, perfect mappings and open mappings (cf. Continuous mapping; Perfect mapping; Open mapping). However, the consideration of decomposition spaces and the "diagram" properties of quotient mappings mentioned above assure the class of quotient mappings of a position as one of the most important classes of mappings in topology. This class contains all surjective, continuous, open or closed mappings (cf. Closed mapping). Quotient mappings play a vital role in the classification of spaces by the method of mappings. Thus, $k$-spaces are characterized as quotient spaces (that is, images under quotient mappings) of locally compact Hausdorff spaces, and sequential spaces are precisely the quotient spaces of metric spaces.

The majority of topological properties are not preserved under quotient mappings. Thus, a quotient space of a metric space need not be a Hausdorff space, and a quotient space of a separable metric space need not have a countable base. Therefore the question of the behaviour of topological properties under quotient mappings usually arises under additional restrictions on the pre-images of points or on the image space. It is known, for example, that if a compactum is homeomorphic to a decomposition space of a separable metric space, then the compactum is metrizable. Under a quotient mapping of a separable metric space on a regular $T_1$-space with the first axiom of countability, the image is metrizable. But there are topological invariants that are stable relative to any quotient mapping. These include, for example, sequentiality and an upper bound on tightness. In topological algebra quotient mappings that are at the same time algebra homeomorphisms often have much more structure than in general topology. Thus, an algebraic homomorphism of one topological group onto another that is a quotient mapping is necessarily an open mapping. Thanks to this, the range of topological properties preserved by quotient homomorphisms is rather broad (it includes, for example, metrizability).

References

[1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
[2]	N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)

Comments

The decomposition space is also called the quotient space.

Properties preserved by quotient mappings (or by open mappings, bi-quotient mappings, etc.) are surveyed in [a2]. However, every topological space is an open quotient of a paracompact regular space, [a1] (cf. also Paracompact space).

References

[a1]	J. Isbell, "A note on complete closure algebras" Math. Systems Theory , 3 (1969) pp. 310–312
[a2]	E.A. Michael, "A quintuple quotient quest" Gen. Topol. Appl. , 2 (1972) pp. 91–138
[a3]	R. Engelking, "General topology" , Heldermann (1989)