**Proof plan.** The argument unfolds the definition of the inductive [limit](/page/Limit) [topology](/page/Topology) in both directions. The forward direction uses the composition of continuous maps. The reverse direction uses the characterisation of [open sets](/page/Open%20Set) in the inductive limit: a subset of $V$ is open if and only if its preimage under each canonical injection $j_n$ is open in $V_n$. **Step 1: Forward direction — [continuity](/page/Continuity) of $T$ implies continuity of each $T \circ j_n$.** Each canonical injection $j_n : (V_n, \tau_n) \to (V, \tau_{\mathrm{ind}})$ is continuous by the definition of the inductive limit topology. If $T : (V, \tau_{\mathrm{ind}}) \to (W, \sigma)$ is continuous, then the composition $T \circ j_n : (V_n, \tau_n) \to (W, \sigma)$ is continuous as a composition of continuous maps. **Step 2: Reverse direction — continuity of each $T \circ j_n$ implies continuity of $T$.** [claim: Preimage Characterisation] Suppose $T \circ j_n$ is continuous for every $n \in \mathbb{N}$. Then for every $\sigma$-open set $U \subseteq W$, the preimage $T^{-1}(U)$ is open in the inductive limit topology on $V$. [/claim] [proof] Let $U \subseteq W$ be $\sigma$-open. For each $n \in \mathbb{N}$, \begin{align*} j_n^{-1}(T^{-1}(U)) &= (T \circ j_n)^{-1}(U), \end{align*} which is open in $(V_n, \tau_n)$ by the continuity hypothesis on $T \circ j_n$. By the characterisation of the inductive limit topology, a subset $A \subseteq V$ is open in $\tau_{\mathrm{ind}}$ if and only if $j_n^{-1}(A)$ is open in $(V_n, \tau_n)$ for every $n \in \mathbb{N}$. Since $j_n^{-1}(T^{-1}(U))$ is open in $(V_n, \tau_n)$ for every $n$, the [set](/page/Set) $T^{-1}(U)$ is open in $(V, \tau_{\mathrm{ind}})$. [/proof] Since $T^{-1}(U)$ is open in $V$ for every $\sigma$-open $U \subseteq W$, the map $T$ is continuous.