How Can a Vertex Represent a 0-Cell and an Edge Represent a 1-Cell in Topology?

[s_jubeh]

Definition: (open cell). Let X be a Hausdorff space. A set c ⊂ X is an open
k − cell if it is homeomorphic to the interior of the open k-dimensional ball
D^k = {x ∈ R^k | x < 1}. The number k is unique by the invariance of
domain theorem, and is called dimension of c.
A 0-cell, 1-cell, 2-cell and 3-cell are called a vertex, edge, face and volume
respectively.

I am confused, what is the meaning of 0-cell and 1-cell. I can imagine a circle and a sphere without borders which resemble 2-cell and 3-cell. But how is vertex and lines are homeomorphic to D⁰ and D¹ respectively. and how is the vertex is 0-cell and edge is 1-cell. I simply can not imagine that.

 

[quasar987]

By definition, R^0 is the point {0}. So a 0-cell is {0}. If you plug k=1 in the definition of open k-dimensional ball, you get that D^1= {x ∈ R | |x| < 1}. That's the interval (-1,1).

 

[s_jubeh]

By definition, R^0 is the point {0}. So a 0-cell is {0}. If you plug k=1 in the definition of open k-dimensional ball, you get that D^1= {x ∈ R | |x| < 1}. That's the interval (-1,1).

Thank you, I get it