Editing Subspace topology (section)

{{Short description|Inherited topology}}
{{Redirect|Induced topology|the topology generated by a family of functions|Initial topology}}

In [[topology]] and related areas of [[mathematics]], a '''subspace''' of a [[topological space]] (''X'', ''𝜏'') is a [[subset]] ''S'' of ''X'' which is equipped with a [[Topological space#Definitions|topology]] induced from that of ''𝜏'' called the '''subspace topology'''<ref name=ttd>{{citation
 | last = tom Dieck | first = Tammo
 | doi = 10.4171/048
 | isbn = 978-3-03719-048-7
 | mr = 2456045
 | page = 5
 | publisher = European Mathematical Society (EMS), Zürich
 | series = EMS Textbooks in Mathematics
 | title = Algebraic topology
 | url = https://books.google.com/books?id=ruSqmB7LWOcC&pg=PA5
 | year = 2008| volume = 7
 }}</ref> (or the '''relative topology''',<ref name=ttd/> or the '''induced topology''',<ref name=ttd/> or the '''trace topology''').<ref>{{citation
 | last = Pinoli | first = Jean-Charles
 | contribution = The Geometric and Topological Framework
 | date = June 2014
 | doi = 10.1002/9781118984574.ch26
 | isbn = 9781118984574
 | pages = 57–69
 | publisher = Wiley
 | title = Mathematical Foundations of Image Processing and Analysis 2}}; see Section 26.2.4. Submanifolds, p. 59</ref>