1. Call a set of integers "good" if and only if there exist integers $a, c \in T, b \notin T$, and satisfy $a<b<c$. Find the number of good subsets of the set $\{1,2, \cdots, 2019\}$.
1. The set $\{1,2, \cdots, 2019\}$ has $2^{2019}$ subsets. Now consider how many of these subsets do not satisfy the condition of being a good subset, i.e., how many subsets are not good.

Obviously, a good subset is neither an empty set nor a singleton set. Therefore, there are at least $1+2019=2020$ subsets that are not good.

Let $T$ be a subset containing at least two elements that is not good, and let the smallest and largest elements in $T$ be $m$ and $M$, respectively.

If there exists an integer $k$ such that $m<k<M$ and $k \notin T$, then $a=m, b=k, c=M$ satisfy the condition of a good subset. Hence, $T$ is good, which is a contradiction.

Thus, all integers $m, m+1, \cdots, M$ must be in the set $T$. This indicates that the subsets of $\{1,2, \cdots, 2019\}$ that are not good are those consisting of at least two consecutive integers. Since any such subset is uniquely determined by the integer pair $(m, M)$, there are $\mathrm{C}_{2019}^{2}$ such subsets.

In summary, the number of good subsets of $\{1,2, \cdots, 2019\}$ is $2^{2019}-2020-\mathrm{C}_{2019}^{2}=2^{2019}-2039191$.