How to extract a filename from a URL & append a word to it?

You can use urllib.parse.urlparse with os.path.basename:

import os
from urllib.parse import urlparse

url = "http://photographs.500px.com/kyle/09-09-201315-47-571378756077.jpg"
a = urlparse(url)
print(a.path)                    # Output: /kyle/09-09-201315-47-571378756077.jpg
print(os.path.basename(a.path))  # Output: 09-09-201315-47-571378756077.jpg

os.path.basename(url)

Why try harder?

In [1]: os.path.basename("https://example.com/file.html")
Out[1]: 'file.html'

In [2]: os.path.basename("https://example.com/file")
Out[2]: 'file'

In [3]: os.path.basename("https://example.com/")
Out[3]: ''

In [4]: os.path.basename("https://example.com")
Out[4]: 'example.com'

Note 2020-12-20

Nobody has thus far provided a complete solution.

A URL can contain a ?[query-string] and/or a #[fragment Identifier] (but only in that order: ref)

In [1]: from os import path

In [2]: def get_filename(url):
   ...:     fragment_removed = url.split("#")[0]  # keep to left of first #
   ...:     query_string_removed = fragment_removed.split("?")[0]
   ...:     scheme_removed = query_string_removed.split("://")[-1].split(":")[-1]
   ...:     if scheme_removed.find("/") == -1:
   ...:         return ""
   ...:     return path.basename(scheme_removed)
   ...:

In [3]: get_filename("a.com/b")
Out[3]: 'b'

In [4]: get_filename("a.com/")
Out[4]: ''

In [5]: get_filename("https://a.com/")
Out[5]: ''

In [6]: get_filename("https://a.com/b")
Out[6]: 'b'

In [7]: get_filename("https://a.com/b?c=d#e")
Out[7]: 'b'

filename = url[url.rfind("/")+1:]
filename_small = filename.replace(".", "_small.")

maybe use ".jpg" in the last case since a . can also be in the filename.