chapter 3 quotient spaces of topological spaces

CHAPTER 3
QUOTIENT SPACES
OF TOPOLOGICAL SPACES
3.1 QUOTIENT SPACES
Let X be a non-empty set. A partition of X is a collection of nonempty
disjoint subsets of X whose union is X. If a is an element of X and R is an
equivalence relation on X then the set { x X : xRa}, denoted by [a], is called
equivalence class of the element a of X relative to the equivalence relation R.
An equivalence relation R on X determines a partition of X and conversely, a
partition of X induces an equivalence relation on X. If R is an equivalence
relation on X, the set of all mutually disjoint equivalence classes in which X is
partitioned relatively to the equivalence relation R, is called the quotient set of X
modulo R and is denoted by X/R. Given a subset A of X , we shall denote the set
{ y : yRx for some x A} of all points which are R-relatives of points of A by
R[A]. In other words, R[A] = {S X/R : S A }. In particular, R[{x}]
( written briefly as R[x]) is the equivalence class to which x belongs. Thus R[x]
simply means [x]. The canonical mapping Q : X X/R defined by Q(x) = [x] is
said to be the quotient map or projection map of X onto the quotient set X/R. Let
(X, ) be a topological space and R an equivalence relation on X. The strongest
20

Chapter 3 21
Quotient Spaces of
Topological Spaces
topology for X/R rendering Q continuous on X is called the quotient topology and
X/R with the quotient topology is called the quotient space of the topological
space X . We shall denote the quotient topology on X/R by Q. Notice that if A is
a subset of X then R[A] = Q -1 (Q(A)) and if A X/R, then
Q -1 (A) = { A : A A }.
3.1.1 Let Q : (X, ) (X/R, Q) be the quotient map of the topological space X
onto the quotient space X/R. Then the following statements are equivalent.
(i) Q is an open(closed) mapping.
(ii) If G is open(closed) in X, then R[G] is open(respectively closed).
(iii) If F is a closed(open) subset of X then the union of all members of
X/R which are subsets of F is closed(respectively open).
3.1.2 Let (X, ) be a topological space. If X is compact, connected, locally
Remarks:
connected, separable or Lindelöf, then so is X/R.
(i) If X is countable then X/R may not be countable.
(ii) Separation axioms are not generally preserved. For example, the
quotient space of a Hausdorff space need not be a Hausdorff space.
3.1.3 Let (X, ) be a topological space and let the quotient space (X/R, Q) be
Hausdorff. Then R is a closed subset of the product space X X.
3.1.4 Let (X, ) be a topological space and let (X/R, Q) be the quotient space.
If the quotient map Q is an open map and R is a closed subset of the
product space X X, then (X/R, Q) is a Hausdorff space.

Chapter 3 22
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Topological Spaces
3.1.5 Let (X, ) be a topological space and let D be an upper semi-continuous
decomposition of X and let D have the quotient topology Q. If G is a
-open set containing a member D of D, then Q[G] is a Q-neighbor-
hood of D.
3.1.6 Let (X, ) be a topological space and let D be a decomposition of X.
Then D is upper semi-continuous if and only if the quotient map Q of X
onto D is closed.
3.1.7 Let (X, ) be a topological space and let D be an upper semi-continuous
decomposition of X such that every member of D is a compact subset of
X and let D have the quotient topology Q. If X is (i) Hausdorff, (ii)
regular, (iii) locally compact or (iv) satisfies the second axiom of
countability, then D also has the corresponding properties.
Let H be a subgroup of a group (G, .). Let the equivalence relation R in G
be defined by xRy x -1 y H. Then given x G, the set { y G : yRx} is an
equivalence class which we denote as xH = {xh : h H}; it is known as the left
coset of H in G. Similarly, Hx is known as the right coset of H in G. When xH =
Hx for all x G, H is said to be a normal subgroup. If H is a normal
subgroup of G, the set of all cosets xH is a group known as the quotient group
of G relative to H and is denoted by G/H.

Chapter 3 23
Quotient Spaces of
Topological Spaces
Let X be a vector (or linear) space over the field K and E be a subspace of
X. For any x X, the set x + E = { x + e : e X } is called coset of the element
x with respect to E. We shall denote x + E by xˆ .
3.1.8 For the subspace E of the linear space X over the field K, the following
statements are equivalent :
(i) u x + E (ii) u – x E (iii) x u + E
3.1.9 Let E be a subspace of a linear space X over the field K. Then
x + E = y + E x – y E and thus (x + z) + E = x + E for any z E
3.1.10 The distinct cosets of the subspace E of a linear space X over the field K
form a partition of X.
The result 3.1.9 asserts that there is an equivalence relation “” in X such
that the distinct cosets of E are the equivalence classes with respect to “”. Thus if
xˆ is one of the distinct cosets of E then xˆ is an equivalence class with x as its
representative element. If x , y X be such that xˆ = yˆ then by 3.1.8, we have
x – y E. Two elements x , y of X are said to be equivalent modulo E if x – y E.
This will be denoted by x y(mod E). The set of all the equivalence classes
modulo E is denoted by X/E. Thus
X/E = { xˆ : x X}.

Chapter 3 24
Quotient Spaces of
Topological Spaces
The set X/E can be made into a linear space over the field K under
algebraic operations defined by
(x + E) + (y + E) = (x + y) + E and (x + E) = x + E
The linear space X/E over the field K is called the quotient space of X modulo E.
elementary.
The following results regarding the quotient space of X modulo E are
3.1.11 dim(X/E) = dim X – dim E ;
3.1.12 The quotient mapping Q : X X/E defined by Q(x) = x + E, is a linear
mapping.
3.1.13 The null space (or kernel) of Q : X X/E is exactly the subspace E.
3.1.14 Let E1 and E2 be the subspaces of a linear space X over the field K. E2 is
a complementary subspace of E1, that is, X = E1 E2 if and only if the
restriction of Q to E2 is an isomorphism of E2 onto X/E1.
3.1.15 Let X1 and X2 be linear spaces over the field K. Suppose that f is a linear
mapping of X1 onto X2. If N is the null space of f then X2 is isomorphic
to X1/N.
3.1.16 Let M be a subspace of the linear space X. If M o represents annihilator of
M, then
(a) M o (X/M)* (b) M* X*/M o

Chapter 3 25
Quotient Spaces of
Topological Spaces
3.2 QUOTIENT GROUPS OF TOPOLOGICAL GROUPS
Let Q be the quotient map of a topological group G onto the quotient
group G/H, that is, the mapping which orders to each x G its equivalence class
xH. The finest topology on G/H for which Q is continuous is known as the
quotient topology; we shall denote it by Q.
3.2.1 If H is a subgroup of a topological group G, G/H is the space of left cosets
equipped with the quotient topology, and Q : G G/H is the canonical
mapping, then
(i) Q is an open, continuous mapping.
(ii) For each a G, the neighborhoods of Q(a) are precisely the sets
Q(aV), where V is a neighborhood of the neutral element e of G.
3.2.2 Let N be a normal subgroup of a topological group G,let Q : G G/N be
the canonical mapping, and equip G/N with the quotient group structure
and the quotient topology. Then
(i) G/N is a topological group.
(ii) Q is an open, continuous mapping.
(iii) For each a G, the neighborhoods of Q(a) for the quotient
topology are precisely the sets Q(a)Q(V), where V is a
neighborhood of the neutral element e of G.
(iv) G/N is separated if and only if N is a closed subset of G.
3.2.3 If G is any topological group and N = { e }, then G/N is a separated
topological group.

Chapter 3 26
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Topological Spaces
3.2.4 Let H be a subgroup of a topological group G and let G/H be the space of
left cosets. Then
(i) G/H is separated if and only if H is closed in G.
(ii) G/H is discrete if and only if H is open in G.
3.2.5 If G is any topological group and C is the connected component of the
neutral element e, then
(i) C is a closed normal subgroup of G.
(ii) For each a G the connected component of a is aC = Ca.
(iii) The quotient topological group G/C is separated and totally
disconnected (i.e., its connected components are all singletons).
3.2.6 Let G and H be topological groups, let f : G H be a continuous
homomorphism of G onto H, and assume that G is compact and H is
separated. Then
(i) H is compact ;
(ii) The kernel N of f is a closed normal subgroup of G, and
(iii) The mapping xN f(x) is a bicontinuous isomorphism of G/N
onto H.
3.2.7 Let G and H be topological groups, let f : G H be a continuous
homomorphism of G onto H with kernel N, then the following conditions
are equivalent:
(i) f is an open mapping.
(ii) For every neighborhood V of eG, f (V) is a neighborhood of eH ;

Chapter 3 27
Quotient Spaces of
Topological Spaces
(iii) The natural isomorphism of G/N onto H is bicontinuous.
3.2.8 Let H be a subgroup of a topological group G. (X, ) be a topological
space and let D be an upper semi-continuous decomposition of X such
that every member of D is a compact subset of X and let D have the
quotient topology Q. If X is (i) Hausdorff, (ii) regular, (iii) locally
compact or (iv) satisfies the second axiom of countability, then D also
has
3.2.9 If G is a metrizable topological group and if N is a closed normal
subgroup of G, then the quotient topological group G/N is metrizable.
3.2.10 Let X be a commutative lattice group, and let E be a subgroup of X that is
an order-convex sublattice. Then X/E is a lattice, and for x, y in X, we
have Q(x y) = Q(x) Q(y) , where Q is the projection of X onto X/E.
3.3 QUOTIENT SPACES OF TOPOLOGICAL LINEAR SPACES
Let (X, ) be a topological vector(or linear) space over the field K , let E
be a subspace of X , and let Q be the quotient map of X onto X/E, that is, the
mapping which orders to each x X its equivalence class ^ x = x + E. Obviously,
Q is linear. The topology on X/E is the quotient topology Q, that is, the finest
topology on X/E for which Q is continuous. Thus the open sets in X/E are the sets
Q(A) such that A + E is open in X; since G + M is open in X whenever G is ,
Q(G) is open in X/E for every open subset G of X; hence Q is an open map.

Chapter 3 28
Quotient Spaces of
Topological Spaces
3.3.1 Let E be a linear subspace of a topological linear space X over K, X/E is
Remarks:
the quotient vector space, Q : X X/E the canonical mapping , and
equip X/E with the quotient topology. Then
(i) X/E is a topological linear space over K ;
(ii) Q is an open , continuous mapping
(iii) For each x X, the neighborhoods of Q(x) = x + E are precisely
the sets Q(x + V) = Q(x) + Q(V), where V is a neighborhood of
X.
(iv) X/E is separated if and only if E is a closed linear subspace of X.
(i) Since the closure of a subspace E of a topological linear space X is
a subspace of X which is also closed, therefore X/ – E is a Hausdorff
topological linear space. The space X/ – E is called the Hausdorff
topological linear space associated with X.
(ii) Let E be a subspace of a topological linear space X. A set is
bounded in E it is bounded as a subset of X. A bounded subset of X/E
may not be the canonical image of a bounded set in X.
3.3.2 Let (X, ) be a topological linear space and let E be subspace of X. Then
Q(B) is a -neighborhood base in X/E for every -neighborhood base B
in X.

Chapter 3 29
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3.3.3 Let (X, ) be a topological linear space and let E be subspace of X. Then
the quotient topology Q on X/E is translation-invariant.
3.3.4 Let (X, ) be a topological linear space over the field K and let E be a
subspace of X. If B is a -neighborhood base in X which possesses the following
properties then Q(B) satisfies the same properties.
(i) For each BB, there exists ß B such that ß + ß B.
(ii) Every B B is radial and circled.
(iii) There exists K, 0 < < 1, such that B B implies BB.
3.3.5 Let (X, ) be a topological linear space over the field K, let E be a
subspace of X, and let Q be the quotient topology on X/E. Then
(X/E, Q) is a topological linear space over K.
3.3.6 If X is a topological linear space and if E is a subspace of X, then X/E is
a Hausdorff space if and only if E is closed in X.
3.3.7 Let the topological linear space X be the algebraic direct sum of its
subspaces M and N. Then X is the topological direct sum of M and N if
and only if the mapping f which orders to each equivalence class modulo
M its unique representative in N is an isomorphism of the topological
linear space X/M onto the topological linear space N.