Quotient topology

Note: Motivation for quotient topology may be useful

Definition

If

$$(X,\mathcal{J})$$ is a topological space, $$A$$ is a set, and $$p:(X,\mathcal{J})\rightarrow A$$ is a surjective map then there exists exactly one topology $$\mathcal{J}_Q$$ relative to which $$p$$ is a quotient map. This is the quotient topology induced by $$p$$

[Expand]

The quotient topology is actually a topology

---

---

TODO: Easy enough

---

Quotient map

Let $(X,\mathcal{J})$ and $(Y,\mathcal{K})$ be topological spaces and let $p:X\rightarrow Y$ be a surjective map.

$p$ is a quotient map if we have

$$U\in\mathcal{K}\iff p^{-1}(U)\in\mathcal{J}$$

Stronger than continuity

If we had $\mathcal{K}=\{\emptyset,Y\}$ then $p$ is automatically continuous (as it is surjective), the point is that $\mathcal{K}$ is the largest topology we can define on $Y$ such that $p$ is continuous

---

TODO: Now we can explore the characteristic property (with $\text{Id}:\tfrac{X}{\sim}\rightarrow\tfrac{X}{\sim}$ ) for now

---