I'm trying to better understand this indicator function detailed on page 8 of the following paper: https://arxiv.org/pdf/1301.1466.pdf
Let $T_k(Y) = \boldsymbol{1}\{|Y| = k+2, \beta_k(C(Y,1)) = 1\}$ where $\beta_k$ gives the $k$th betti number and $C(Y,1)$ is the Cech complex on the collection of balls radius $1$ with centres given by the points in $Y$. So essentially $T_k(Y)$ is $1$ if $C(Y,1)$ is a minimal k-dimensional cycle.
Understanding the technical details of what the above means is not necessary to answering my question. To better understand this object I'd like to do a simple computation: $$\int_{(\mathbb{R}^2)^2} T_1(0,\boldsymbol{y})d\boldsymbol{y}$$ I really have almost no idea when this indicator function is or is not 1. (the motivation for doing this computation comes from theorem 2.2 in the linked paper)
If you require anything explained in further detail please ask!