---- > [!definition] Definition. ([[initial topology]]) > Let $X$ be a set and $\{(X_{i}, \tau_{i})\}_{i \in I}$ a family of [[topological space|topological spaces]]. Suppose $\{ f_{i}: X \to X_{i} \}_{i \in I}$ is an indexed set of functions $X \to X_{i}$. The **initial topology $\tau_{0}$ on $X$ with respect to $\{ f_{i} \}$** is the [[comparable topologies|coarsest]] [[topological space|topology]] on $X$ making all $f_{i}$ [[continuous]]. If $\Gamma$ denotes the set of all topologies $\tau$ on $X$ such that each $f_{i}$ is [[continuous]], then $\tau_{0}$ manifestly equals their [[intersection of topologies is a topology|meet]]: $\tau_{0}=\bigcap_{\tau \in \Gamma} \tau.$ > The term **coarse topology** are also used. Sometimes 'weak topology' is used, but we try to avoid overloading that term. > > > The initial topology has as [[subbasis for a topology|subbasis]] the collection $\mathscr{S}= \{ f_{i} ^{-1} (U) : U \in \tau_{i}, i \in I \}.$ > It therefore is [[topology generated by a basis|generated by]] the [[basis for a topology|basis]] $\mathscr{B}=\{\bigcap_{k=1}^{n} f_{i_{k}} ^{-1}(U_{k}): n \in \mathbb{N}, U_{k} \in \tau_{i_{k}}\}.$ > - [ ] category theory ([[categorical limit|limit]]) perspective; though tbh i don't find it adds a bunch of insight at this time > [!basicproperties] > - (*Convergence*) A [[sequence]] $(x_{n})$ converges to $x$ in $(X, \tau_{0})$ if and only if the sequences $f_{i}(x_{n})$ [[converge]] to $f_{i}(x)$ for all $i \in I$.[^1] In this case the notation $x_{n} \rightharpoonup x$ is sometimes used. > > > > [!proof]- Proof. > > Suppose $(x_{n}) \to x$. Fix $i \in I$. By construction, $f_{i}$ is [[continuous]], [[the sequential continuity lemma|so]] it preserves the limit: $f_{i}(x_{n}) \to f_{i}(x)$. > > > > Conversely, suppose $f_{i}(x_{n}) \to f_{i}(x)$ for all $i \in I$. WTS $(x_{n}) \to x$. Fix a basic neighborhood $B$ of $x$; we know $B=\bigcap_{k=1}^{n}f_{i_{k}}^{-1}(U_{k})$ for some $n \in \mathbb{N}$ and $U_{k} \in \tau_{i_{k}}$. There exist $N_{i_{k}} \in \mathbb{N}$ past which $f_{i_{k}}(x_{n})$ lives in $U_{k}$. Letting $N=\max \{ N_{i_{k}} \}$, we have $x_{n}$ in $B$ past $n$, as required. > > > > > [^1]: That the [[product topology]] is the [[product topology|topology of pointwise convergence]] follows from this fact. > [!basicexample] > - [[product topology]] > - [[subspace topology]] > - [[weak topology]] > - [[weak topology|weak-star topology]] > - [[weak convergence of measures]] > - [[convergence in distribution of random variables]] > - etc.... ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```