This is possible to do if you give up the requirement that your topology is T$_1$, and it has already been done, with quite a few papers. Google khalimsky line or digital topology, I will include some reference and comments a bit later.
The general idea is that you declare each odd integer to be an open set (as in the usual topology) but the even integers are not open (though as usual each closed). Each even integer $2n$ has a minimal neighborhood consisting of itself together with the two neighboring odd integers, so called Khalimsky line. This makes the set of all integers connected, and so called COTS (connected ordered topological space) have been studied. The square of the Khalimsky line is the Khalimsky plane, and there is even a Jordan curve theorem for the Khalimsky plane.
Here is just one (early) paper in this area (you could find many more,
for that matter, not just google, but a search on MSE website returns many results for digital topology).
Kong, T. Yung; Kopperman, Ralph; Meyer, Paul R.
A topological approach to digital topology.
Amer. Math. Monthly 98 (1991), no. 10, 901–917.
It can be downloaded at http://www.jstor.org/stable/2324147
Also, Digital Topology, Azriel Rosenfeld
The American Mathematical Monthly
vol. 86, No. 8 (Oct., 1979), pp. 621-630
The Khalimsky Line as a Foundation for Digital Topology
Erik Melin, Connectedness and continuity in digital spaces with the Khalimsky topology