BUG: read_csv not applying dtype to index col when dtype is globally specified

[gaow]

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Reproducible Example

# Not expected behavior: index are still int
import pandas as pd
from io import StringIO
data = "1,a\n2,b"
df = pd.read_csv(StringIO(data), index_col=0, dtype=str, header=None)
df.index

Issue Description

This is separated from #44632. More discussions on #9435. In version 1.4.0 it does fix the case when dtype is defined for a column specifically:

df = pd.read_csv(StringIO(data), index_col=0, dtype={0:str}, header=None)

works. But not when you specify dtype globally.

Expected Behavior

I expect by specifying index_col=0, dtype=str which requires all columns be str, then the first column is also a str and is used as index as well. With current version of pandas I am not sure what's the alternative implementation for my expected behavior.

Installed Versions

INSTALLED VERSIONS

commit : bb1f651
python : 3.8.5.final.0
python-bits : 64
OS : Linux
OS-release : 5.4.0-39-generic
Version : #43-Ubuntu SMP Fri Jun 19 10:28:31 UTC 2020
machine : x86_64
processor : x86_64
byteorder : little
LC_ALL : en_US.UTF-8
LANG : en_US.UTF-8
LOCALE : en_US.UTF-8

pandas : 1.4.0
numpy : 1.19.5
pytz : 2020.5
dateutil : 2.8.1
pip : 20.2.4
setuptools : 50.3.1.post20201107
Cython : None
pytest : 6.2.2
hypothesis : None
sphinx : 4.3.1
blosc : None
feather : 0.4.1
xlsxwriter : None
lxml.etree : 4.6.2
html5lib : None
pymysql : None
psycopg2 : None
jinja2 : 2.11.2
IPython : 7.19.0
pandas_datareader: None
bs4 : 4.9.3
bottleneck : None
fastparquet : None
fsspec : 0.8.5
gcsfs : None
matplotlib : 3.3.4
numba : None
numexpr : 2.7.2
odfpy : None
openpyxl : 3.0.9
pandas_gbq : None
pyarrow : 0.17.1
pyreadstat : None
pyxlsb : None
s3fs : None
scipy : 1.6.0
sqlalchemy : 1.3.22
tables : None
tabulate : 0.8.7
xarray : 0.16.2
xlrd : 2.0.1
xlwt : None
zstandard : 0.15.1

[phofl]

As mentioned on the pr, my preference would be to keep as is. But open for discussion

[Marcos-C7]

I experienced this problem because I was thinking that dtype globally specified was global indeed. I agree with gaow that global should be global, unless the implementation brings more problems than solutions.

[DriesSchaumont]

This issue also confused me today. The docs seem to say:
"Use str or object together with suitable na_values settings to preserve and not interpret dtype"

But using str or object, for the dtype parameter in read_csv causes pandas to still tries to infer a suitable dtype for the index. I am dealing with a fixed number of index columns and a variable number of data columns, in which case I cannot seem to stop pandas inferring any of the columns. If I specify only the index columns in a dict, the dtype of the data columns are inferred, and if I use str, the index columns dtypes are inferred. I tried to use defaultdict, but that does not seem to work either.