# Characteristic properties for topological pushouts and pullbacks

So far in my topology class we've talked about several topological constructions (namely the subspace topology, the quotient topology, the (finite and infinite) product topology, and the disjoint union topology). In each case we had some sort of "characteristic property" to associate to each. For instance, the characteristic property of the subspace topology is

If $A$ is a subspace of $X$, and $T$ is another topological space, then a function $f:T \rightarrow A$ is continuous $\iff$ $i_A \circ f$ is continuous, where $i_A:A \rightarrow X$ is the inclusion function.

Another example is for the product topology:

If $X,Y,T$ are topological spaces, then the functions $f:T \rightarrow X$, $g:T\rightarrow Y$ are continuous $\iff$ $(f,g): T \rightarrow X \times Y$ is continuous (the projection functions lurk in the background).

I would like to know if there is a characteristic property for the pullback and the pushout of topological spaces, and if it is in the form of the statements above (namely in the form of an "if and only if" statement that gives a condition for detecting the continuity of certain maps).

I have some idea of what the maps that come with spaces are (e.g., I think in the case of pushouts, the two associated maps are maps that are compositions: the injection of a set into a disjoint union, composed with a quotient map into the disjoint union modulo an equivalence relation), however I'm not sure what the corresponding properties are.

Thanks in advance for any help! Please let me know if the question is imprecise or unclear.

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Pushouts and pullbacks can be defined in a general category, so you might be interested in looking at the category theoretic definition. – Ncat Nov 15 '13 at 4:20
Ah yes! I took a look at the Wikipedia articles for both of them (in the general category theory setting). I'm having difficulty translating the universal properties given there into statements of the form that I'm looking for. – Bachmaninoff Nov 15 '13 at 4:25
There is a category of topologies, Top, where the objects are topological spaces and the morphisms are continuous maps. So if you just make the necessary terminology replacements, that should get you going in the right direction. – Ncat Nov 15 '13 at 4:28
I think I'm getting stuck with thinking about what the maps I "care about" are. For the subspace and product topologies, I "care" about maps into the subspace/product. Similarly, for the quotient and disjoint union topologies, I "care" about functions out of the quotient/disjoint union...I suspect I'm supposed to look at functions into the pushout/functions out of the pullback? I'll try to formulate an answer to this after some sleep. Thanks for the help! – Bachmaninoff Nov 15 '13 at 4:50
It's exactly the reverse, actually. – Zhen Lin Nov 15 '13 at 11:37