Unique smallest and largest topology ideas What does it mean to be a smallest topology of a set $X$. I would guess that it would be a topology of $X$ which has least number of elements and similarly for largest topology it would have to be largest number of elements? Am I correct? This looks fairly straightforward unless I am missing something
For example:
If I am understanding this correctly if $X=\{a,b,c\}$ and $\mathcal{T_1}=\{\emptyset, X, \{a,b\}, \{a\}\}$ and
$\mathcal{T_2}=\{\emptyset, X, \{b,c\}, \{a\}\}$. Then the largest topology will be  $\mathcal{T_l}=\{\emptyset, X, \{a,b\}, \{a\}, \{b\}, \{b,c\}\}$ because you can get $\mathcal{T_1}$ and $\mathcal{T_2}$ from it and the smallest will be   $\mathcal{T_s}=\{\emptyset, X, \{a\}\}$
 A: We tend not to use the terms 'smallest' or 'largest' topology, but finest and coarsest (see wikipedia for a definition). Some people do use smallest and largest, but it's less common and in my opinion more ambiguous.
In this sense, the discrete topology (every subset is open) is the finest topology, and the coarsest is the indiscrete (trivial - only $X$ and $\emptyset$ are open) topology.
In terms of their cardinality, the indiscrete topology is the 'smallest' topology on $X$ and the discrete topology is the 'largest' (with the caveat that for infinite sets, the discrete topology has the same cardinality as some coarser topologies on the set, and so is not the unique largest topology).

With regard to your edit, you may be thinking about the infimum and supremum of topologies with respect to the partial order given by subset inclusion. To expand on this a little, the set of all topologies on a set $X$ form a partially ordered set with the partial order given by subset inclusion $\subseteq$. And so we can formally say that a topology $\tau$ on $X$ is less than or equal to another topology $\tau'$ if $\tau\subseteq\tau'$, equivalently if $\tau'$ is a finer topology than $\tau$. Let's denote this partially ordered set $(P_X,\subseteq)$ where $P_X$ is the set of all topologies on $X$.
$(P_X,\subseteq)$ has a unique maximal element given by the discrete topology, and a unique minimal element given by the indiscrete topology. We can also consider the collection of elements $A=\{\tau_\lambda\mid \lambda\in\Lambda\}$ and ask if there exists an element $\tau\in P_X$ such that $\tau_\lambda\subseteq\tau$ for all $\lambda\in\Lambda$, and for any other $\tau'\in P_X$ with this property $\tau\subseteq\tau'$. It turns out that such an element exists and is in fact unique (existence is not always guaranteed of supremums in general partially ordered sets, although uniqueness is). We call $\tau$ the supremum (sometimes called the least upper bound) of $A$ and write $\tau=\sup A$. It is the coarsest topology which contains all open sets from the $\tau_{\lambda}$. In your example, $\mathcal{T}_l=\sup\{\mathcal{T}_1,\mathcal{T}_2\}$.
We can similarly define the infimum (sometimes called the greatest lower bound) $\tau^*=\inf A$ which is the finest topology which is contained in all $\tau_{\lambda}$. In your example $\mathcal{T}_s=\inf\{\mathcal{T}_1,\mathcal{T}_2\}$.
We can find the infimum $\inf A$ by simply taking the intersection of all the topologies in $A$, that is $$\inf A=\bigcap A=\bigcap_{\lambda\in\Lambda}\tau_{\lambda}$$ and it's not hard to see that this is in $P_X$ and it is the infimum of $A$. It is not the case that the supremum is the union of the topologies in $A$ as the union is not in general a topology, however the union does form a subbase of the supremum.
