IntervalIndex is_non_overlapping_monotonic is incorrect when endpoints are shared

[jschendel]

Code Sample, a copy-pastable example if possible

In [2]: idx = pd.IntervalIndex.from_breaks(range(4), closed='both')

In [3]: idx
Out[3]:
IntervalIndex([[0, 1], [1, 2], [2, 3]]
              closed='both',
              dtype='interval[int64]')

In [4]: idx.is_non_overlapping_monotonic
Out[4]: True

Problem description

The intervals have an overlap in the endpoints, 1 is shared among [0, 1], [1, 2] and 2 is shared among [1, 2], [2, 3], but is_non_overlapping_monotonic returns True indicating that the intervals are non-overlapping.

Expected Output

I'd expected is_non_overlapping_monotonic to return False.

If my interpretation of "overlapping" is incorrect, and in this context means an interval overlap (i.e. not a point/has positive length), then it'd be nice if this were more explicitly stated.

Output of pd.show_versions()

INSTALLED VERSIONS

commit: None
python: 3.5.2.final.0
python-bits: 64
OS: Windows
OS-release: 10
machine: AMD64
processor: Intel64 Family 6 Model 78 Stepping 3, GenuineIntel
byteorder: little
LC_ALL: None
LANG: None
LOCALE: None.None

pandas: 0.20.1
pytest: 3.0.7
pip: 9.0.1
setuptools: 34.4.1
Cython: 0.25.2
numpy: 1.13.0rc2
scipy: 0.18.1
xarray: 0.9.5
IPython: 5.3.0
sphinx: None
patsy: 0.4.1
dateutil: 2.6.0
pytz: 2017.2
blosc: 1.4.4
bottleneck: 1.2.0
tables: 3.4.2
numexpr: 2.6.1
feather: None
matplotlib: 2.0.2
openpyxl: 2.4.0
xlrd: 1.0.0
xlwt: None
xlsxwriter: None
lxml: 3.7.3
bs4: None
html5lib: None
sqlalchemy: 1.1.10
pymysql: None
psycopg2: None
jinja2: 2.9.6
s3fs: None
pandas_gbq: 0.1.6
pandas_datareader: 0.4.0

[jschendel]

At first glance, I think this amounts to using strict inequalities when closed='both', and using the existing inclusive inequality code for is_non_overlapping_monotonic otherwise.

[jreback]

closed='both' is not very well tested. Almost all things exercise closed='right', which is the default. So for certain could add some tests.