TITLE: Direct limit of topological spaces
QUESTION [10 upvotes]: Let $X$ be a topological space.
Suppose $X_n$ are subspaces of X with $X_1 \subset X_2 \subset \cdots \subset X$.
I want to prove that $\varinjlim X_n =\cup_n X_n$.
I have some trouble in proving that $\cup_n X_n$ satisfies the universal property of direct limits.
More precisely, I have a difficulty to prove the map $\Phi : \cup_n X_n \longrightarrow Y$ by $x \longrightarrow j_n(x)$ if $x\in X_n$ is continuous, where $Y$ is any topological space with continuous maps $j_n :X_n \longrightarrow Y $.

REPLY [16 votes]: Take $X=\mathbb{N}$ with the cofinite topology and $X_i = \{1,\dotsc,i\}$. Then each $X_i$ carries the discrete topology, and it follows easily that their colimit $\varinjlim_i X_i$ also carries the discrete topology. But the union $\cup_i X_i$, equipped with the subspace topology, actually equals $X$ and doesn't carry the discrete topology.