(that now you seem to have deleted after my comment https://en.m.wikipedia.org/wiki/Topology_(disambiguation) )
"Topology is a branch of mathematics concerned with geometric properties preserved under continuous deformation (stretching without tearing or gluing)"
That is indeed the established meaning of topology, more so in mathematics and the blog post was on applied mathematics. That it may mean something else in other contexts is irrelevant.
I rest my case.
> The most common uses of "topology", whenever used to convey a geometry-related idea, is in the more general sense meaning "surfaces"
Erm, citation please.
I included a search on Amazon on topology https://www.amazon.com/s?k=Topology&sprefix=topology+%2Caps%... (without even adding the keyword maths. None of the results seem to be about surfaces. Shouldn't there have been a few ? Wouldn't Amazon search results reflect the general sense meaning ?).
If it were true, wouldn't the Wikipedia pages have talked about that general sense meaning first ?
Alternatively, I would say, take a breath. Is this hill really the one worth dying on ? There are better ones. Have a good day and if work permits, get yourself a juicy topology book, it can be interesting, if presented well.