2014 08 29 quotient spaces
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Quotient spaces 1 / 10
Quotient spaces
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Quotient spaces j Denition 2 / 10
1 Denition
2 Maps going out of quotients
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Quotient spaces j Denition 2 / 10
Quotient maps
Denition (Quotient map)Let X; Y be topological spaces and p: X! Y a surjectivemap. Then p is called a quotient map if U Y is open ifand only if p 1(U) is open in X.
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Quotient spaces j Denition 2 / 10
Quotient maps
Denition (Quotient map)Let X; Y be topological spaces and p: X! Y a surjectivemap. Then p is called a quotient map if U Y is open ifand only if p 1(U) is open in X.
Remarks
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Quotient spaces j Denition 2 / 10
Quotient maps
Denition (Quotient map)
Let X; Y be topological spaces and p: X! Y a surjectivemap. Then p is called a quotient map if U Y is open ifand only if p 1(U) is open in X.
Remarks
p is a quotient map if and only if p is surjective and
p 1(A) is closed in X if and only if A Y closed.
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Quotient spaces j Denition 2 / 10
Quotient maps
Denition (Quotient map)
Let X; Y be topological spaces and p: X! Y a surjectivemap. Then p is called a quotient map if U Y is open ifand only if p 1(U) is open in X.
Remarks
p is a quotient map if and only if p is surjective and
p 1(A) is closed in X if and only if A Y closed.
Maps which are open, continuous and surjective arequotient maps.
Q ti t j D iti 2 / 10
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Quotient spaces j Denition 2 / 10
Quotient maps
Denition (Quotient map)
Let X; Y be topological spaces and p: X! Y a surjectivemap. Then p is called a quotient map if U Y is open ifand only if p 1(U) is open in X.
Remarks
p is a quotient map if and only if p is surjective and
p 1(A) is closed in X if and only if A Y closed.
Maps which are open, continuous and surjective arequotient maps.
Maps which are closed, continuous and surjective are
quotient maps.
Q ti t j D iti 3 / 10
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Quotient spaces j Denition 3 / 10
Saturated subsets
Denition (Fiber)Let p: X! Y be a surjective map between sets andy2 Y. The inverse image set
p`1(fyg) = fx2 X j p(x) = yg
will be denoted as p 1(y) and referred to as the ber of y.
Quotient spaces j Denition 3 / 10
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Quotient spaces j Denition 3 / 10
Saturated subsets
Denition (Fiber)Let p: X! Y be a surjective map between sets andy2 Y. The inverse image set
p`1(fyg) = fx2 X j p(x) = yg
will be denoted as p 1(y) and referred to as the ber of y.
Denition (Saturated set)
Let p: X! Y be a surjective map between sets. We saythat A X is saturated (with respect to p) ifA \ p 1(y) 6= ; implies p 1(y) A
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Quotient spaces j Denition 4 / 10
Remarks
Quotient spaces j Denition 4 / 10
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Quotient spaces j Denition 4 / 10
Remarks
A is saturated if and only if A = p 1(p(A)).
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Quotient spaces j Denition 4 / 10
Remarks
A is saturated if and only if A = p 1(p(A)).
If U Y, then p 1(U) is saturated.
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Q o e p e j e o /
Remarks
A is saturated if and only if A = p 1(p(A)).
If U Y, then p 1(U) is saturated.
p is a quotient map if and only if p is surjective,continuous, and A saturated and open implies p(A) open.
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Remarks
A is saturated if and only if A = p 1(p(A)).
If U Y, then p 1(U) is saturated.
p is a quotient map if and only if p is surjective,continuous, and A saturated and open implies p(A) open.
p is a quotient map if and only if p is surjective,
continuous, and A saturated and closed implies p(A)closed.
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Quotient topology
Quotient topologyLet X be a space, A a set, and p: X! A a surjective map.Then there is exactly one topology on A that makes p a
quotient map, such topology is called quotient topology.
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Quotient topology
Quotient topologyLet X be a space, A a set, and p: X! A a surjective map.Then there is exactly one topology on A that makes p a
quotient map, such topology is called quotient topology.
Remark
One must check that:
fi = fU A j p 1(U) is open in Xg
is a topology on A.
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Partitions
Denition (Partition)
Let X be a set. A partition X of X is a collection ofdisjoint subsets of X with union X
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Partitions
Denition (Partition)
Let X be a set. A partition X of X is a collection ofdisjoint subsets of X with union X
Quotient space
Let X be a topological space, X be a partition of X, andp: X! X the natural surjection. If we give X the quotient
topology induced by p, the space X is called a quotientspace of X
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Quotients and subspaces
If p: X! Y is a quotient map and A X is a subspace, itdoes not necessarily follow that the restriction of p:
A ! p(A) is a quotient map. However we have:
Quotient spaces j Denition 7 / 10
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Quotients and subspaces
If p: X! Y is a quotient map and A X is a subspace, itdoes not necessarily follow that the restriction of p:
A ! p(A) is a quotient map. However we have:
Theorem (Quotients and subspaces)
Let p: X! Y be a quotient map and A X a subspace,that is saturated with respect to p, and let q: A ! p(A)the restriction of p. Then:
Quotient spaces j Denition 7 / 10
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Quotients and subspaces
If p: X! Y is a quotient map and A X is a subspace, itdoes not necessarily follow that the restriction of p:
A ! p(A) is a quotient map. However we have:
Theorem (Quotients and subspaces)
Let p: X! Y be a quotient map and A X a subspace,that is saturated with respect to p, and let q: A ! p(A)the restriction of p. Then:
if A is open or closed, then q is a quotient map,
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Quotients and subspaces
If p: X! Y is a quotient map and A X is a subspace, itdoes not necessarily follow that the restriction of p:
A ! p(A) is a quotient map. However we have:
Theorem (Quotients and subspaces)
Let p: X! Y be a quotient map and A X a subspace,that is saturated with respect to p, and let q: A ! p(A)the restriction of p. Then:
if A is open or closed, then q is a quotient map,
if p is open or closed, then q is a quotient map.
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Quotients and compositions
Theorem
Composition of quotient maps is a quotient map.
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Fundamental theorem
Theorem (Fundamental theorem)
Let p: X! Y be a quotient map. Let Z be a space, andg: X! Z a continuous map that is constant on each berof p. Then g induces a continuous map g: Y! Z such
that g p= g.
X Z
Y
g
p
g
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d h
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Fundamental theorem
Theorem (Fundamental theorem, for equivalencerelations)
Let X be a space, and an equivalence relation on X.Let g: X! Z be a continuous map such that g(x) = g(x0)whenever x x0. Then, if we denote by p: X! X= thequotient map, the map g induces a continuous map
g: X=! Z such that g p= g
X Z
X=
g
pg