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.
Best Answer
HINTS:
a. Show that if each projection is continuous, then the basic open sets in the usual definition of the product topology must all be open; this is completely trivial if you understand the definition of the product topology.
b. This is also just a matter of checking the definitions: if $\mathrm{id}\upharpoonright W:W\to X$ is continuous, then for every open set $U$ in $X$, $\left(\mathrm{id}\upharpoonright W\right)^{-1}[U]$ must be open in $W$. What is another description of the set $\left(\mathrm{id}\upharpoonright W\right)^{-1}[U]$, one that does not mention any function? Is it true that the collection of all of these inverse images is a topology on $W$?
c. The projections from $\Bbb R^n$ to $\Bbb R$ are linear functionals; use (a).