$\mathfrak{Top}$ and injective objects My question is very simple. Let $\mathfrak{Top}$ be the category of all topological spaces and continuous functions between them. Does such category have enough injectives? Is there a simple way to define injective objects of $\mathfrak{Top}$ with additional hypothesis on the topology? 
 A: I suppose you mean injective objects with respect to the class of all monomorphisms. Recall that a monomorphism in $\mathbf{Top}$ is precisely an injective continuous map. Thus:

Lemma. Every non-empty indiscrete space is injective.
Proof. Let $Z$ be non-empty and indiscrete. Then every map $X \to Z$ is automatically continuous. Thus, given any injective continuous map $f : X \to Y$ and any map $g : X \to Z$, there is at least one map $h : Y \to Z$ such that $h \circ f = g$, so $Z$ is indeed an injective object.

It follows that $\mathbf{Top}$ has enough injective objects, because every topological space admits an injective continuous map into a non-empty indiscrete space. 
In particular, the injective objects in $\mathbf{Top}$ are precisely the non-empty indiscrete spaces: indeed, for any $X$, there is a non-empty indiscrete space $Z$ and an injective continuous map $f : X \to Z$, so if $X$ is injective, there is also a continuous map $r : Z \to X$ such that $r \circ f = \mathrm{id}_X$, and it is not hard to see that this implies $X$ is non-empty and indiscrete.
