$\mathrm{S}\_{\mathrm{N}}1$ reaction involves only one molecule in the rate determining step. So, the molecule which undergoes $\mathrm{S}\_{\mathrm{N}}1$ reaction should be stable when it forms a positive ion. $\ce{(CH3)3C^+}$ is considered stable due to inductive effect from the three carbon atoms and also hyper-conjugation. But, what about bridged compounds like 1-bromobicyclo[2.2.1]heptane:  

![1-bromobicyclo[2.2.1]heptane](https://i.stack.imgur.com/PpsMV.png) 

When $\ce{Br^-}$ is abstracted the positive ion formed will be stable and therefore this will undergo $\mathrm{S}\_{\mathrm{N}}1$ reaction. Am I thinking right? Please help me correct myself.