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Pick out the true statements.

a. Let $\{X_i:i\in I\}$ be topological spaces. Then the product topology is the smallest topology on $X = \prod X_i$ such that each of the canonical projections $\pi : X \to X_i$ is continuous.

b. Let $X$ be a topological space and $W\subseteq X$. Then, the induced subspace topology on $W$ is the smallest topology such that $\mathrm{id}\upharpoonright W : W\to X$, where $\mathrm{id}$ is the identity map, is continuous.

c. Let $X =\Bbb R^n$ with the usual topology. This is the smallest topology such that all linear functionals on $X$ are continuous.

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    What, if anything, have you tried? Also, on this site we'd prefer that you not use the imperative ("Pick out..."). It'd ruffle fewer feathers to begin with something like "I can't figure out which of these is true. For (a) I tried but I got stuck at and I haven't got a clue about how to approach (b) and (c)."2012-09-23

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