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Point-set topology
Definition. A topology on a set X is a collection τ = {U ⊂ X} of subsets satisfying the
following conditions:
(1) ∅, X ∈ τ ;
(2) the union of any family of sets of τ belongs to τ ;
(3) the intersection of a finitely many sets of τ belongs to τ .
The sets of τ are called open sets. The complementary sets X \ U to open sets are called closed
sets. A set (X, τ ) together with topology is called topological space.
Examples.
(1) The number line R with the collection of arbitrary unions of intervals U =
S
(aα , bα ) and
α
empty set ∅ and whole R as open sets.
(2) The plane R2 . A set U ⊂ R2 is open if and only if for each point x ∈ U it contains a
sufficiently small open disc centered at x.
(3) An arbitrary set X with τ = {∅, X}. This topology is called trivial.
(4) An arbitrary set X with all its subsets as open sets. This topology is called discrete.
(5) X = R (or C) and an open subset U consists of non-zero points of some polynomial P (x):
U = {x ∈ R|P (x) 6= 0}. This topology is called Zarisski topology.
Definition. Let Y ⊂ X be a subset of a topological space X. Then Y inherits a topology
whose open sets the intersections open sets of X with Y . This topology on Y is called induced
topology from X.
Definition. Let X be a topological space with an equivalence relation ∼ which is reflexive
(x ∼ x), symmetric (x ∼ y implies y ∼ x), and transitive (x ∼ y and y ∼ z imply x ∼ z). The set
of equivalence classes Y := X/ ∼ can be endowed with quotient topology whose open sets are the
subsets of Y which preimages under the canonical projection X → X/ ∼ are open sets in X.
Examples.
(1) Let X = I := [0, 1] be the unit interval with the
equivalence relation, 0 ∼ 1, gluing the end-points.
The quotient space X/ ∼ is a circle S 1 .
0
1
(2) Let X = I 2 := [0, 1] × [0, 1] be the
unit square with the equivalence relation,
(t, 0) ∼ (t, 1) for all 0 6 t 6 1, gluing the
opposite sides. The quotient space X/ ∼
is a cylinder.
(3)
Let X = I 2 be the unit square with the
equivalence relation, (t, 0) ∼ (1 − t, 1) for
all 0 6 t 6 1, gluing the opposite sides
reversing the orientation. The quotient
space X/ ∼ is the Möbius band.
(4) Let X = I 2 be the unit square
with the equivalence relation,
(t, 0) ∼ (t, 1) and (0, t) ∼ (1, t)
for all 0 6 t 6 1, gluing the opposite sides in pairs. The quotient
space X/ ∼ is the torus.
(5) Let X = I 2 be the unit square with the equivalence relation, (t, 0) ∼ (t, 1) and (0, t) ∼
(1, 1 − t) for all 0 6 t 6 1, gluing the opposite sides in pairs and reversing the orientation
of one pair. The quotient space X/ ∼ is the Klein bottle which cannot be embedded into
3-space R3 without self-intersection.
On the standard picture of the Klein bottle the set of self-intersection is represented by
a circle where the tube hits itself and goes through to the interior of itself. If we cut the
Klein bottle along the black contour we will get two Möbius bands. So one may think
about the Klein bottle as two Möbius bands glued together along their boundary circles.
(6) Let X = I 2 be the unit square with the equivalence relation, (t, 0) ∼ (1 − t, 1) and
(0, t) ∼ (1, 1 − t) for all 0 6 t 6 1, gluing the opposite sides in pairs and reversing the
orientation of both pairs. The quotient space X/ ∼ is the projective plane. It also cannot
be embedded into R3 without self-intersection.
The standard picture is called the cross cap. The set of
self-intersections is an interval. Its neighborhood is shown
on the right picture.
Definition. A topological space X is said to be Hausdorff if for any two distinct points
x, y ∈ X there are disjoint open sets U ∋ x and V ∋ y: U ∩ V = ∅.
Definition. A topological space X is said to be compact if for any system of open sets covering
X there is a finite subsystem still covering X.
Examples.
(1) (Heine—Borel—Lebesgue) A closed interval [0, 1] is compact in the topology induced from
R (see example (1) above).
(2) A semi-closed interval (0, 1] is not compact. An example of a cover without finite subcover
∞
S
(1/n, 1].
is (0, 1] =
n=2
(3) One-point compactification X̄ of a topological space X is obtained by adding one extra
point ∞ (called a point at infinity). Its open sets are the open sets of X and the sets of
the form U ∪ {∞}, where U ⊂ X is an open subset of X such that X \ U is compact. The
space X̄ is always compact.
Definitions.
(1) A map f : X → Y of the topological spaces is called continuous at the point x0 ∈ X if
for any open set V ∋ f (x0 ) in Y there is an open set U ∋ x0 such that f (U ) ⊂ V .
(2) A map f : X → Y of the topological spaces is called continuous (everywhere) if the
pereimage f −1 (V ) ⊂ X of any open set V ⊂ Y is open in X.
Theorem. (Weierstrass) Any continuous function on a compact topological space attains its
maximum and minimum.
Definition. A homeomorphism f : X → Y of the topological spaces X and Y is a a continuous
1-to-1 map such that its inverse f −1 is also continuous.
Exercises.
(1) Prove that the open interval (0, 1) is homeomorphic to the whole line R.
(2) Prove that the open interval (0, 1) is not homeomorphic to the closed interval [0, 1].
(3) Prove that the circle in Example (1) obtained by gluing the end-points of the interval is
homeomorphic to the circle S 1 := {(x, y) ∈ R2 |x2 + y 2 = 1} with the topology induced
from R2 .
(4) Prove that the one-point compactification of R is homeomorphic to the circle S 1 .
(5) Prove that the circle S 1 is not homeomorphic to the figure 8, also known as the wedge of
two circles S 1 ∨ S 1 .
(6) Prove that the cylinder and the annulus 1 6 x2 + y 2 6 2 are homeomorphic.
(7) Prove that the cylinder is not homeomorphic to the Möbius band.
(8) Prove that the spheres of different radii are homeomorphic.
(9) Prove that the one-point compactification of R2
(x, y, z)
is homeomorphic to the sphere S 2 := {(x, y, z) ∈
R3 |x2 + y 2 + z 2 = 1}. Hint: Use the stereographic
y
x
, 1−z
) from S 2 to R2
projection (x, y, z) 7→ ( 1−z
shown on the right picture.
y
x
(10) Prove that a sphere is not homeomorphic to a torus.
( 1−z
, 1−z
)
Definitions.
(1) An n-dimensional topological manifold is a second countable1 Hausdorff topological space
each point of which has an open neighborhood homeomorphic to an open ball {(x1 , . . . , xn ) ∈
Rn |x21 + · · · + x2n < 1} of Rn .
(2) An n-dimensional topological manifold with boundary is a second countable Hausdorff
topological space each point of which has a neighborhood homeomorphic either to an
open ball of Rn or to a half-ball {(x1 , . . . , xn ) ∈ Rn |x21 + · · · + x2n < 1, xn > 0}.
Jordan Curve Theorem. A simple closed plane curve, that is a homeomorphic image of S 1
into the plane R2 , separates the plane into two parts one of which is homeomorphic to a disc.
1This means that there is a countable collection of the open sets such that any open set can be represented as
a union of sets of this collection.