# Why is topology called 位相幾何学?

Topology in English is called 位相幾何学 in Japanese; also, topological space is called 位相空間. But why is topology called 位相幾何学? What is the correspondence between `topo` and `位相`? What is the origin of `位相`?

• There might not be a direct correspondence. I don't see one between topo- and 位相, nor between -logy and 幾何学. Good question, though!
– user1478
Apr 2, 2013 at 13:20
• Not related to your specific question, but I really like this question about the origin of mathematical terminology in Japanese (in this case the origin of "functor").
– Earthliŋ
Apr 2, 2013 at 13:44
• @snailplane Although the subject is called 位相幾何学, the concept of a topology (as in collection of open sets) is just called 位相 in Japanese. So we really need a correspondence 位相 <=> topology.
– Earthliŋ
Apr 3, 2013 at 14:04
• Recall that in English, names of disciplines often stand for the objects they study, where as in Japanese, they have different names. Thus topology can be translated in Japanese 位相 or 位相幾何学, depending on the context. To make things more complicated, topology usually stands for application of topological notions to geometry, not a study of topological spaces themselves (which is usually called general topology or point-set topology), which is why the word 幾何 comes into play. Aug 16, 2013 at 15:12

The correspondence isn't direct; if 位相幾何学 were loan translated into English it would be 位相 (topological) 幾何学 (geometry).

Interestingly, though, 位相 means phase (i.e. of a sinusoidal function) as well as topology, and that means that the term 位相空間 is ambiguous between phase space (in physics) and topological space (in mathematics).

EDIT: To clarify the etymology:

The first morpheme in topology is Gk. τόπος 'place', which corresponds to 位 (position). I would suggest that 相 in this context means 'aspect', 'behaviour', and not 'mutual' (its other sense). In Chinese, the two senses of 相 have different readings (xiang4 vs xiang1). Putting it all together, the term 位相 refers to the disposition of place -- that is, how the place is arranged; in other words, its topology.

• But where does the term 位相 come from?
– ssb
Apr 2, 2013 at 14:33
• A bit of googling reveals that there are other usages of the word 位相, such as this I'm curious where the word came from, too. Apr 2, 2013 at 15:01
• The first morpheme in topology is Gk. τόπος 'place', which corresponds to 位 (position). I would suggest that 相 in this context means 'aspect', 'behaviour', and not 'mutual' (its other sense). In Chinese, the two senses of 相 have different readings (xiang4 vs xiang1). Putting it all together, the term 位相 refers to the disposition of place -- that is, how the place is arranged; in other words, its topology. Apr 3, 2013 at 2:38
• Thanks, but there is one point remained unclear to me:what is 'position' in topology? A topological space merely defined by 'what is open set', whereas 'position' seems need to be defined in vector space... Apr 3, 2013 at 15:15
• @Popopo: Euler referred to early topological concepts as constituting a geometria situs (geometry of place) as opposed to physical geometry which accounts for lengths and distances, and the Japanese is a precise calque of this. Apr 4, 2013 at 0:44