The idea behind a binary adjunction space is fairly simple – we take two topological spaces, sayX andY, and specify the parts which are to be glued to each other. Formally, (binary) adjunction spaces are defined as follows.
Definition 3.1.LetX, Y be topological spaces andA⊆X. Given a continu- ous functionf :A→Y, we define the adjunction space ofX andY underf as:
X∪fY :=X⊔Y∼
where ∼is the reflexive transitive closure of the relation that identifies (a,1) and(f(a),2)for eacha∈A. The topology onX∪fY induced from the quotient
of the disjoint union topology onX⊔Y is called the adjunct topology, and we will denote this topology by τA.
The construction ofX∪fY can be described with the following diagram,
A X Y X⊔Y X∪fY f idX ϕ1 φX ϕ2 φY q
where the maps φX and φY are simply the compositions of the canonical
inclusions ϕi and the quotient map q associated to the equivalence relation ∼, i.e.φX =q◦ϕ1 andφY =q◦ϕ2. Throughout this thesis we will refer to
maps such asφX andφY ascanonical maps. Observe that since the maps ϕi
and q are continuous, so are the canonical mapsφX and φY. We will follow
standard practice and abbreviate the above diagram to the following:
A X
Y X∪fY idX
f φX
φY
The diagram above is commutative, which means that on the subsetA, the maps φX and φY ◦f are equal. Observe also that the points of X∪f Y are
equivalence classes, and be described explicitly as:
• [x,1] ={(x,1)} ifx /∈A
• [a,1] ={(f(a),2)} ∪ {(a′,1)|a′∈f−1(f(a))} for eacha∈A • [y,2] ={(y,2)} ∪ {(a,1)|a∈f−1(y)}for eachy∈f(A)
We also have the following observation, which follows immediately from the definition of the topologyτA.
Proposition 3.2.Let U be a subset of X∪f Y. ThenU is open in X∪fY
iffφ−X1(U)andφ−Y1(U)are open in their respective spaces.
The following lemma shows that the adjunction spaceX∪fY possesses a
certain universal property.
Lemma 3.3.LetZbe a topological space whereψX:X →Z andψY :Y →Z
are continuous maps such thatψX(a) =ψY ◦f(a)for eacha∈A. Then there
is a unique continuous mapg:X∪fY →Z that makes the following diagram
commute. A X Y X∪fY Z f idX φX ψX φY ψY g
Proof. We defineg by:
g([x, i]) = (
ψX(x) ifi= 1
ψY(x) ifi= 2
Observe first that this is well-defined: if [x,1] = [y,2], thenf(x) =yand thus:
g([x,1]) =ψX(x) =ψY ◦f(x) =ψY(y) =g([y,2]),
We now show that this map is continuous. Let U ⊆Z be open and consider the preimage g−1(U). According to Prop. 3.2, it suffices to show that both
φ−X1(g−1(U)) and φ−1 Y (g
−1(U)) are open in their respective spaces. It is not
hard to see that φ−X1(g−1(U)) = ψ−1
X (U), which is open in X since we as-
sumed ψX was continuous. A similar situation holds for Y, from which we
can conclude thatg−1(U) is open inX∪
fY, and thusg is continuous.
To see that g is unique, suppose we have some other continuous map g′:X∪
fY →Zsuch thatψX =g′◦φXandψY =g′◦φY. Consider an element
[x,1] inX∪fY. Theng([x,1]) =g◦φX(x) =ψX(x) =g′◦φX(x) =g′([x,1])
and thus g([x,1]) =g′([x,1]) for all x ∈X. The argument for Y is similar,
It will be useful to know under what circumstancesX andY topologically embed into X ∪f Y under the canonical maps φX and φY.2 The following
lemma summarises a number of results.
Lemma 3.4 (Basic facts about the canonical maps).Let X∪fY be an
adjunction space.
1.φY is always an injection.
2. Iff is an injection, then so isφX.
3. IfA is open, thenφY is an open map.
4. Iff is injective and open map, thenφX is open.
5.φY is always a topological embedding.
6. Iff is an injective, open map, thenφX is a topological embedding.
Proof. See Appendix A1. ⊓⊔
In the above result we have omitted certain well-known results for the case when Ais assumed to be closed, since these are not useful for our purposes. The following is an immediate corollary of the above lemma, and is a useful identification of some sufficient conditions required to turn the canonical maps φX andφY into open embeddings.
Corollary 3.5.IfA is an open subset ofX andf is an injective, open map, thenφX andφY are both open topological embeddings.
It will also be useful to know which properties the adjunction spaceX∪fY
naturally inherits fromX and Y. The following lemma collects some results regarding the preservation of various properties, the proofs of which can be found in the Appendices.
Lemma 3.6.Let X∪fY be an adjunction space.
1. If BX and BY are bases for X and Y respectively, and φ
X and φY are
open maps, then the collection
B={φX(U)|U ∈ BX} ∪ {φY(V)|V ∈ BY}
forms a basis for the topologyτA.
2. IfX andY are connected andAis non-empty, thenX∪fY is connected.
3. IfX andY are compact, then so isX∪fY.