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Let $T_a$ be a family of topologies on a set $X$. What is the smallest topology containing all the $T_a$? Obviously, the smallest it could be is the union of all the $T_a$, but that's not always a topology. So is it just the topology generated by the subbasis $\bigcup T_a$? I feel like this is too simple of an answer, since the question is phrased asking to prove that there is a unique smallest topology containing all the $T_a$.

(I'm working through Munkres' Topology on my own.)

A l'Maeaux
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    One crisp way to prove uniqueness is to consider that the topology on subbasis $\cup T_a$ is the intersection of all the topologies containing all the $T_a$. – Kevin Carlson Jan 07 '14 at 22:47

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Yes, it's the topology that has $\bigcup T_a$ as a subbasis. It is also the final topology with respect to the family $\iota_a\colon (X,T_a) \to X$, where $\iota_a$ is the identity map for all $a$.

I feel like this is too simple of an answer, since the question is phrased asking to prove that there is a unique smallest topology containing all the $T_a$.

It is that simple, though you need to say a word or two about the fact that it is indeed coarser than any topology containing all $T_a$ (and hence it's uniquely determined as the smallest element in a partially ordered set; not only a minimal element, which need not be unique). But a few words really suffice.

Daniel Fischer
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As Daniel wrote it is the topology that has $\bigcup T_a$ as a subbasis. So it is the set of all unions of finite intersections of sets in $\bigcup T_a$. Another way to describe it is as the intersection of all topologies on $X$ containing $\bigcup T_a$ (there is at least one, namely the power set $\mathcal P(X)$ ). However, we can make use of the knowledge that each $T_a$ is a topology, because this requires us to only intersect sets from different $T_a$'s. For example, if you have $T_1$ and $T_2$, then the smallest topology containing them is the set of unions of sets of the form $U_1\cap U_2$, where $U_1\in T_1$ and $U_2\in T_2$. In particular, if $T$ is any topology and $T_2$ is the topology $\{\emptyset, X, A\}$, for any subset $A$ in $X$, then the smallest topology is called $T(A)$ and it can be expressed as $$T(A)=\{U\cup(V\cap A)\mid U\in T, V\in T\}$$

Stefan Hamcke
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  • I would add that this definition of $\bigcup T_{\alpha}$ uses sub-basis as defined by Wikipedia's (primary) definition. The alternative definition presented by Wikipedia is Munkres' definition of basis. It is important to point this out, because Munkres' definition of sub-basis does not respect the equivalence you put forth: that $\bigcup T_{\alpha}$ is both the set of all unions of finite intersections and the intersection of all topologies on $X$ containing $\bigcup T_{\alpha}$ (which is true using Wikipedia's primary definition). See https://en.wikipedia.org/wiki/Subbase – Muno May 22 '17 at 12:49