Definition 3.1.7 (Intial Topology).label Let $X$ be a set, $\curl{(Y_i,\tau_i)}_{i\in I}$ be a family of topological spaces, and $\curl{f_i:X\to Y_i}_{i\in I}$ be a family of maps then there exists a topology $\tau$ on $X$ satisfying:
- (1)
$f_{i}\in C(X;Y_{i})$ for each $i\in I$
- (U)
$\tau$ is the coarsest topology on $X$ such that all $f_{i}$ are continuous
- (U’)
For any topological space $Z$ and map $g:Z\to X$ we have $g$ is continuous if and only if $f_{i}\circ g:Z\to Y_{i}$ is continuous for each $i\in I$.
- (4)
A subbase for $\tau$ is given by
\begin{align*}\cali{S}=\curl{f_i\inv(U_i):U_i\in\tau_i,i\in i}\end{align*} - (5)
(1)+(U)$\iff$(U’)
called the intial/weak topology on $X$ generated by the maps $\curl{f_i}_{i\in I}$.
Proof. Let $\tau\define \tau(\cali{S})$ then by Definition 3.1.6
- (1)
$\curl{f_i\inv(U_i):U_i\in\tau_i,i\in I}\suf \tau$ hence $f_{i}\in C((X,\tau);Y_{i})$ for each $i\in I$
- (U)
If $\tau'$ is a topology on $X$ satisfying (1) then $\curl{f_i\inv(U_i):U_i\in\tau_i,i\in I}\suf \tau$ hence $\cali{S}\suf \tau'$ and so $\tau\suf\tau'$
- (U’)
If $g$ is continuous then $f_{i}\circ g$ is continuous for each $i\in I$ since composition of continuous functions is continuous. Since finite intersections of subbasis sets form a basis and preimages commute with intersections, it suffices to check continuity of $g$ on a subbasis. Let $i\in I$ and $V\suf Y_{i}$ open then
\begin{align*}g\inv(f_{i}\inv(V))=(f_{i}\circ g)\inv(V)\end{align*}is open in $Z$ by continuity of $f_{i}\circ g$ as desired.
- (4)
Clear
- (5)
Clear
$\square$
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