Why topological embedding continuous? Problem
Why do we require a topological embedding to be continuous? Compared to other categories we want a space to be isomorphic to some subspace. Translating this to topological spaces that is saying there is a homeomorphism to a subset in the subspace topology and considering this map as with codomain of the ambient space that doesn't need to be continuous: $X\cong B\leq Y$
Idea
Is it true that for: $f:X\to Y\text{ continuous}$ and $g:Y\to X\text{ continuous}$
$f\text{ has continuous left inverse}\iff f\text{ is topological embedding}$
$g\text{ has continuous right inverse}\iff g\text{ is quotient map}$
 A: If $i:A\to(X,\tau)$ is injective, then we can equip $A$ with the initial topology for $i$, that is the family $\{i^{-1}(U)\mid U\in\tau\}$. This is the coarsest topology making $i$ continuous, and a function $f$ from a space to $A$ is continuous iff $i\circ f$ is continuous. I particular, if $A\subset X$, and $i(a)=a$, then this gives the usual subspace topology $\{U\cap A\mid U\in\tau\}$. A map $f:Y\to X$ such that $f(Y)\subseteq A$ is now continuous iff its restriction $f':Y\to A$ is continuous, as $f=i\circ f'$.
A left inverse is also called a retraction, a right inverse is also called a section.
Note that if $i:A\to X$ is an embedding, and $r:X\to A$ is a retraction, then $A\cong i(A)\subseteq X$ and $ir:X\to X$ sends every point in $X$ to a point in $i(A)$ and is constant on $i(A)$. So the categorical retractions are equivalent to the retractions as they are usually introduced in topology.
An example of an embedding without a retraction is $\{0\}\times I\cup X\times\{0\}\subset X\times I$, where $X=\{0,1/n\mid n\in\Bbb N\}$.
A quotient map without a section could be the quotient map 
$$q:I\to\{[0,1/2],(1/2,1]\},\qquad q(x)=A\iff x\in A$$
On the other hand, every section is an embedding and every retraction is a quotient map.
