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<p>functions continuous.</p>

<p><big>Definition</big></p>
<p>Given a set  and an <a href="page.php?w=indexed_family">indexed family</a>  of <a href="page.php?w=topological_space">topological space</a>s with functions</p>

<p>the initial topology  on  is the <a href="page.php?w=coarsest_topology">coarsest topology</a> on  such that each</p>

<p>is <a href="page.php?w=continuous_function_%28topology%29">continuous</a>.</p>

<p><b>Definition in terms of open sets</b></p>

<p>If  is a family of topologies  indexed by  then the [[least upper bound]] topology</p><p>
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