Initial topology

Remarks

The preimages $f_i^{-1}(V)$ make each $f_i$ continuous, and any topology that makes each $f_i$ continuous must contain them; this gives the equivalence of the two forms. Universal property: for a topological space $Z$, a map $g \colon Z \to X$ is continuous for the initial topology if and only if $f_i \circ g$ is continuous for every $i$.

Example: for a subset $X$ of a space $Y$ and the inclusion $\iota \colon X \to Y$, the initial topology is the subspace topology. Its open sets are the sets $\iota^{-1}(V) = X \cap V$ with $V$ open in $Y$. Example: for a product $X = \prod_i Y_i$ and the projections, the initial topology is the product topology.

Non-example: the discrete topology on $\R$ makes the identity map to $\R$ with its usual topology continuous, but it is not the initial topology for this map. It is strictly finer than needed: the initial topology here is the usual topology, in which $\set{0}$ is not open.

The initial topology is also called the weak topology or the induced topology for the family.