Simplicial Structures in Topology

I.1 Topology 5
Let X be a topological space with topology U. A subset A ⊂ X canonically inherits
a topology from A, namely, the induced topology in A whose open sets are the
intersections of the open sets of X with A:
B = {U ∩ A |U ∈ A}.
(I.1.4) Lemma. Let X be a topological space with topology U, Y be a set, and
q: X → Y be a function. The set of subsets with anti-images open in X
is a topology on Y.
V = {U ⊂ Y | q −1 (U) ∈ U}
Proof. We need to verify axioms A1, A2, and A3.
A1: q−1 (/0)=/0 ∈ U, q−1 (Y)=X ∈ U.
A2: It holds because
q −1
� �
�
= �
q −1 (Uα).
Uα
α∈J
A3: Likewise,
q −1
� �
n� n�
Ui = q
i=1 i=1
−1 (Ui). �
When the function q is a surjection, the topology V of Y, defined above, is called
the quotient topology on Y induced by q. We note that the set Y could be given by a
partition of X in disjoint classes whose union is precisely X. To make this fact clear,
we now give three examples.
(I.1.5) Example. Let D2 = {(x,y) ∈ R2 | x2 + y2 ≤ 1} be the two-dimensional unit
disk with boundary S1 = {(x,y) ∈ R2 | x2 + y2 = 1} (one-dimensional sphere). Let
D2 ≡ be the set whose elements are
α∈J
1. {(x,y)} for the points (x,y) ∈ D 2 such that x 2 + y 2 < 1
2. {(x,y),(−x,−y)} for the boundary points (x,y) ∈ S 1
in this case, we say that the boundary points (x,y) and (−x,−y) are being identified,
as in Fig. I.2. In this way, we obtain the topological space D 2 ≡, with the quotient
topology induced by the epimorphism (that is to say, by the surjective function)
q: D 2 → D 2 ≡; this is the real projective plane RP 2 . We shall return to this meaningful
example in Sect. I.3.
(I.1.6) Example. Consider I 2 = I × I (I is the unit interval [0,1]) with the product
topology. Let us now take the set I 2 ≡ whose elements are the following sets (see also
Fig. I.3):
1. {(x,y)},if0< x < 1, 0 < y < 1
2. {(x,0),(x,1)},if0< x < 1