- Quotient space
In
topology and related areas ofmathematics , a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by anequivalence relation . This is commonly done in order to construct new spaces from given ones.Definition
Suppose "X" is a
topological space and ~ is anequivalence relation on "X". We define a topology on thequotient set "X"/~ (the set consisting of allequivalence class es of ~) as follows: a set of equivalence classes in "X"/~ is open if and only if their union is open in "X".This is the quotient topology on the quotient set "X"/~.Equivalently, the quotient topology can be characterized in the following manner: Let "q" : "X" → "X"/~ be the projection map which sends each element of "X" to its equivalence class. Then the quotient topology on "X"/~ is the
finest topology for which "q" is continuous.Given a
surjective map "f" : "X" → "Y" from a topological space "X" to a set "Y" we can define the quotient topology on "Y" as the finest topology for which "f" is continuous. This is equivalent to saying that a subset "V" ⊆ "Y" is open in "Y" if and only if itspreimage "f"−1("V") is open in "X". The map "f" induces an equivalence relation on "X" by saying "x"1~"x"2if and only if "f"("x"1) = "f"("x"2). The quotient space "X"/~ is thenhomeomorphic to "Y" (with its quotient topology) via the homeomorphism which sends the equivalence class of "x" to "f"("x").In general, a surjective, continuous map "f" : "X" → "Y" is said to be a quotient map if "Y" has the quotient topology determined by "f".
Examples
*Gluing. Often, topologists talk of gluing points together. If "X" is a topological space and points are to be "glued", then what is meant is that we are to consider the quotient space obtained from the equivalence relation "a ~ b" if and only if "a = b" or "a = x, b = y" (or "a = y, b = x"). The two points are henceforth interpreted as one point.
*Consider the unit square "I"2 = [0,1] × [0,1] and the equivalence relation ~ generated by the requirement that all boundary points be equivalent, thus identifying all boundary points to a single equivalence class. Then "I"2/~ is homeomorphic to the unit sphere "S"2.
*Adjunction space . More generally, suppose "X" is a space and "A" is a subspace of "X". One can identify all points in "A" to a single equivalence class and leave points outside of "A" equivalent only to themselves. The resulting quotient space is denoted "X"/"A". The 2-sphere is then homeomorphic to the unit disc with its boundary identified to a single point: "D"2/∂"D"2.
*Consider the set "X" = R of allreal number s with the ordinary topology, and write "x" ~ "y"if and only if "x"−"y" is aninteger . Then the quotient space "X"/~ ishomeomorphic to theunit circle "S"1 via the homeomorphism which sends the equivalence class of "x" to exp(2π"ix").
*A vast generalization of the previous example is the following: Suppose atopological group "G" acts continuously on a space "X". One can form an equivalence relation on "X" by saying points are equivalent if and only if they lie in the same orbit. The quotient space under this relation is called the orbit space, denoted "X"/"G". In the previous example "G" = Z acts on R by translation. The orbit space R/Z is homeomorphic to "S"1."Warning": The notation R/Z is somewhat ambiguous. If Z is understood to be a group acting on R then the quotient is the circle. However, if Z is thought of as a subspace of R, then the quotient is an infinite
bouquet of circles joined at a single point.Properties
Quotient maps "q" : "X" → "Y" are characterized among surjective maps by the following property: if "Z" is any topological space and "f" : "Y" → "Z" is any function, then "f" is continuous if and only if "f" O "q" is continuous.
The quotient space "X"/~ together with the quotient map "q" : "X" → "X"/~ is characterized by the following
universal property : if "g" : "X" → "Z" is a continuous map such that "a"~"b" implies "g"("a")="g"("b") for all "a" and "b" in "X", then there exists a unique continuous map "f" : "X"/~ → "Z" such that "g" = "f" O "q". We say that "g" "descends to the quotient".The continuous maps defined on "X"/~ are therefore precisely those maps which arise from continuous maps defined on "X" that respect the equivalence relation (in the sense that they send equivalent elements to the same image). This criterion is constantly being used when studying quotient spaces.
Given a continuous surjection "f" : "X" → "Y" it is useful to have criteria by which one can determine if "f" is a quotient map. Two sufficient criteria are that "f" be open or closed. Note that these conditions are only sufficient, not necessary. It is easy to construct examples of quotient maps which are neither open nor closed.
Compatibility with other topological notions
* Separation
** In general, quotient spaces are ill-behaved with respect to separation axioms. The separation properties of "X" need not be inherited by "X"/~, and "X"/~ may have separation properties not shared by "X".
** "X"/~ is aT1 space if and only if every equivalence class of ~ is closed in "X".
** If the quotient map is open then "X"/~ is aHausdorff space if and only if ~ is a closed subset of theproduct space "X"×"X".
*Connectedness
** If a space is connected orpath connected , then so are all its quotient spaces.
** A quotient space of asimply connected orcontractible space need not share those properties.
* Compactness
** If a space is compact, then so are all its quotient spaces.
** A quotient space of alocally compact space need not be locally compact.
*Dimension
** Thetopological dimension of a quotient space can be more (as well as less) than the dimension of the original space;space-filling curve s provide such examples.ee also
Topology
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Subspace (topology)
*Product space
*Disjoint union (topology)
*Final topology Algebra
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Quotient group
*Quotient space (linear algebra)
*Quotient category References
* Stephen Willard, "General Topology", (1970) Addison-Wesley Publishing Company, Reading Massachusetts.
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