BUG: `date_range` `inclusive` parameter behavior doesn't match interval notation when `start == end`

[mixilchenko]

Pandas version checks

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• I have confirmed this bug exists on the latest version of pandas.

• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

import pandas as pd

pd.date_range('20230926', '20230926', inclusive='right')
# Result: DatetimeIndex(['2023-09-26'], dtype='datetime64[ns]', freq='D')
# Expected: DatetimeIndex([], dtype='datetime64[ns]', freq='D')

Issue Description

Date ranges of (date, date] and [date, date) equal to [date, date] but not (date, date) which is unintuitive.
Issue #46331 looks related to this

Expected Behavior

import pandas as pd

date = pd.Timestamp('20230926')
r = pd.date_range(date, date, inclusive='right')
l = pd.date_range(date, date, inclusive='left')
n = pd.date_range(date, date, inclusive='neither')
pd.testing.assert_index_equal(r, n)
pd.testing.assert_index_equal(l, n)

Installed Versions

INSTALLED VERSIONS

commit : 89bd569
python : 3.9.10.final.0
python-bits : 64
OS : Darwin
OS-release : 22.2.0
Version : Darwin Kernel Version 22.2.0: Fri Nov 11 02:03:51 PST 2022; root:xnu-8792.61.2~4/RELEASE_ARM64_T6000
machine : arm64
processor : arm
byteorder : little
LC_ALL : None
LANG : ru_RU.UTF-8
LOCALE : ru_RU.UTF-8

pandas : 2.2.0dev0+283.g89bd5695df
numpy : 1.26.0
pytz : 2021.3
dateutil : 2.8.2
setuptools : 58.1.0
pip : 21.2.4
Cython : None
pytest : None
hypothesis : None
sphinx : None
blosc : None
feather : None
xlsxwriter : None
lxml.etree : None
html5lib : None
pymysql : None
psycopg2 : None
jinja2 : 3.0.3
IPython : 8.1.1
pandas_datareader : None
bs4 : None
bottleneck : None
dataframe-api-compat: None
fastparquet : None
fsspec : None
gcsfs : None
matplotlib : 3.6.0
numba : None
numexpr : None
odfpy : None
openpyxl : 3.0.10
pandas_gbq : None
pyarrow : None
pyreadstat : None
python-calamine : None
pyxlsb : None
s3fs : None
scipy : 1.11.1
sqlalchemy : None
tables : None
tabulate : None
xarray : None
xlrd : 2.0.1
zstandard : None
tzdata : 2023.3
qtpy : None
pyqt5 : None

[mixilchenko]

#40245 is also related

[PiotrekB416]

take

[DavidToneian]

Date ranges of (date, date] and [date, date) equal to [date, date] but not (date, date) which is unintuitive.
Issue https://github.com//issues/46331 looks related to this

I'm not sure whether parts of the pandas API would lead one to believe this, but from a mathematical point of view, I find this very intuitive:
(date, date], [date, date), and [date, date] are all intervals containing exactly one value: date.
(date, date), however, is the empty set (interval of all values that are larger than date, but smaller than date).

[PiotrekB416]

From a purely mathematical point of view (date, date] means that the interval contains every x that matches the inequality $date < x \leq date$
and since no x can match that condition the interval should be an empty set

[DavidToneian]

Thanks for the correction, you're right of course. :)

[mickjermsurawong-stripe]

Contractually, this is a regression from earlier version
This is a regression from earlier version of pandas at least in version 1.2.4

import pandas as pd
print(pd.__version__)
date = pd.Timestamp('20230926')
r = pd.date_range(date, date, closed='left')
print(r)
# 1.2.4
# DatetimeIndex([], dtype='datetime64[ns]', freq='D')

Same problem described above in version at least in 1.5.3

import pandas as pd
print(pd.__version__)
date = pd.Timestamp('20230926')
r = pd.date_range(date, date, closed='left')
print(r)
# 1.5.3
# DatetimeIndex(['2023-09-26'], dtype='datetime64[ns]', freq='D')

From the new version, code-wise it is surprising that we are only handling exclusivity for the two ends:

pandas/pandas/core/arrays/datetimes.py

Lines 430 to 432 in 8dab54d

	if start == end:
	if not left_inclusive and not right_inclusive:
	i8values = i8values[1:-1]

The i8values will return 1 element since it satisfies the earlier function of both two-ended inclusivity

pandas/pandas/core/arrays/datetimes.py

Lines 2746 to 2749 in e6d0c1a

	* This method is faster for generating weekdays than dateutil.rrule
	* At least two of (start, end, periods) must be specified.
	* If both start and end are specified, the returned dates will
	satisfy start <= date <= end.

I can't think of the scenario when the timestamp is equal and we will have more than one element.. I think we should be handling exclusivity independently with or instead?

 if start == end: 
     if not left_inclusive or not right_inclusive:
         i8values = []

[tscheburaschka]

All this started with the poorly informed change requested in #43394
Whether this has any relation to the renaming of kwarg closed to inclusive is beyond my insights.
I would strongly support an API coherent with the mathematical definition of intervals.
Quote from https://en.wikipedia.org/wiki/Interval_(mathematics)#Including_or_excluding_endpoints:

Each interval (a, a), [a, a), and (a, a] represents the empty set, whereas [a, a] denotes the singleton set {a}.

Additionally, the exact significance of the periods parameter when combined with inclusive should be clarified. From reading #46331 it seems to me, that the length of the result depends on the value of inclusive which is very counterintuitive to me.

How all this could be done without breaking a lot of peoples code (again) in non-obvious ways is not clear to me. Perhaps it requires a resuscitation of the closed kwarg.
I could help with discussing the pros and cons of various approaches but I don't have time for actual implementations (especially since I have not yet contributed to pandas and there seems to be a substantial initial investment required).