Question 11

Question

What is the initial topology?

Answer

Given a set $X$ and an indexed family $(Y_i{)}_{i\in I}$ of topological spaces with functions
$f_i:X\to Y_i,$
the initial topology $\tau$ on $X$ is the weakest topology on $X$ such that
$f_i:(X,\tau )\to Y_i$
is continuous.

Indexed Family

An indexed family here means like an index in programming, such as 0, 1, 2, 3.
For example here $Y_1,Y_2,Y_3,\dots$