- #1
kakarotyjn
- 98
- 0
Definition. A complex K,when regarded as a topological space,is called a polyhedron and written |K|.
I think it is easy to understand the definition,but there are some theorem and problems involving it confused me.
1.Let K be a simplicial complex in [tex] E^n[/tex],if we take the simplexes of K separately and give their union the identification topology,then we obtain exactly |K|.
And also I don't understand explicitly about identification topology,I only know a set A in Y is open if and only if [tex] \pi ^{ - 1} (A) [/tex] is open in X for which [tex] \pi [/tex]
is the map from X to the identification space Y.
What exactly does it means by saying 'if we take simplexes of K separately and give their union the identification topology'?What occurs to the simplexes of K by this?
2.Check that |CK| and C|K| are homeomorphic spaces.
Does it need proof? I think |CK|=C|K|,isn't it? If not,what's the difference between them?
Thank you very much!![Smile :smile: :smile:](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
I think it is easy to understand the definition,but there are some theorem and problems involving it confused me.
1.Let K be a simplicial complex in [tex] E^n[/tex],if we take the simplexes of K separately and give their union the identification topology,then we obtain exactly |K|.
And also I don't understand explicitly about identification topology,I only know a set A in Y is open if and only if [tex] \pi ^{ - 1} (A) [/tex] is open in X for which [tex] \pi [/tex]
is the map from X to the identification space Y.
What exactly does it means by saying 'if we take simplexes of K separately and give their union the identification topology'?What occurs to the simplexes of K by this?
2.Check that |CK| and C|K| are homeomorphic spaces.
Does it need proof? I think |CK|=C|K|,isn't it? If not,what's the difference between them?
Thank you very much!