# A universal property for the subspace topology

Let $X$ be topological space and $Y$ be a subset of $X$ with $i\colon Y\to X$ the inclusion map. Show that the induced topology of $Y$ is characterized by the following property: A function $f\colon Z \to Y$ of a topological space $Z$ into $Y$ is continuous if and only if $i\circ f$ is continuous.

-
I wasn't sure what "application" meant and I know what the property should be, so I edited the question. Please consider writing about what you've tried, and phrasing the question in such a way that it doesn't sound like you're ordering folks around. [Textbooks are allowed to do this, but I doubt you'd walk into a colleague's office and say exactly what you wrote above; that would be somewhat rude.] Welcome, by the way! – Dylan Moreland Jun 22 '12 at 1:32
@DylanMoreland "application" is French for mapping; a translator's false friend that often trips up French-speaking people on English-language math websites. – user31373 Jun 22 '12 at 1:46
Welcome to math.SE. Since you are new, I want to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. If this is homework, please add the [homework] tag; people will still help, so don't worry. Also, many users find the use of the imperative ("Find", "Show", etc) to be rude when asking for help. Please consider rewriting your post. – Arturo Magidin Jun 22 '12 at 2:36

## 1 Answer

It is also helpful to see this definition of induced topology on a subset as a special case of the notion of initial topology with respect to a set of partial functions $f_\lambda: X \to X_\lambda, \lambda \in \Lambda$, where $X$ is a set, and $X_\lambda, \lambda \in \Lambda$ is a family of topological spaces. This is the topology which has a subbase the sets $f^{-1}_\lambda(U)$ for all open $U$ in $X_\lambda$ and $\lambda \in \Lambda$. It has the universal property that a function $f: Y \to X$, where $Y$ is a topological space, is continuous if and only if $f_\lambda \circ f$ is continuous for all $\lambda$. This is how for example one defines the topology on a manifold when the $f_\lambda$ are a family of charts.

-