I have encountered two properties in different areas of math. One is the property (T) of groups and the other is the property (T) of topologies. What is the connection between these two ? Thank you.
Note: A topology on $\tau_Y$ (where $\tau_Y$ is a topology on $Y$) is said to have property (T) if the set $T=\left\{(V,y)\colon y\in V\right\}$ is open in $\tau_Y\times Y.$ The group case is standard. (please see link) A possible relation should come from the Fell topology.
Clarification: A topology on $Y$ is a collection of subsets of $Y$ . A collection of sets may be thought of as a set of sets or, in this case, as a set of elements whose elements are the open sets in $Y$. Therefore, we can define a topology on this set; namely, we can define a topology on the topology of $Y$ . Now $\tau_Y$ is a topological space (as is $Y$) and we can form the product topology on $\tau_Y\times Y$. The topology on $\tau_Y$ is said to have property (T) if $\{(V,y):y∈V\text{ and }V \text{ open in }Y\}$ is an open subset in the product topology on $\tau_Y\times Y$.