I am taking a class on Elementary Geometric Topology and studying Kinsey's book: Topology of Surfaces. When defining an n-cell, the words "interior","boundary" and "frontier" is used without specifying the ambient space.
I have tried to understand the definition in the following way.
I have assumed, maybe the resulting space will be in $\mathbb{R}^n$ so that she refers that using those words. But then AB line segment would not be a 1 cell since its interior is empty for n>1. On the other hand if the ambient space is assumed to be the cell itself then every point is interior. What am I missing here?
In class, the teacher defined a regular cell complex immediately in the following way:
Let X be a Hausdorff Topological Space. Define a regular cell complex K on X by
0) Let $K_{0}$ be a finite set of points, called vertices or 0-cells in X.
1) Construct $K_1$ by adding a finite number of edges or 1 cells, connecting vertices of $K_0$. The set of points in $K_1-K_0$ is a finite disjoint union of open edges. If e is one of these, then there is a continous map f from the interval [0,1] into X taking (0,1) homeomorphically onto e, taking the endpoints 0,1 to the endpoints of e.
2) Construct $K_2$ by adding a finite number of polygonal discs or 2-cells spanning these edges. The set of points in $K_1-K_0$ is a finite disjoint union of open discs. If e is one of these, then there is a continous map f from the $B^2$ into X taking $D^2$ homeomorphically onto e, and the circle $S^1=B^2-D^2$ homeomorphically onto cl(e)-Int(e).
k) Construct $K_k$ so the set of points in $K_k-K_{k-1}$ is a finite disjoint union of open k-discs or k-cells. If e is one of these, then there is a continous map f from $B^k$ into X taking $D^k=Int(B^k)$ homeomorphically onto e and taking $S^{k-1}=B^k-D^k$ homeomorphically onto cl(e)-Int(e)
In this manner, construct sets
$K_0 \subset K_1 \subset K_2 \subset……\subset K_n …$
so that every point in X lies in a cell in one of the $K_k$s. This is called a regular cell structure on X.
I have also looked at the book Basic Topology by Armstrong, there it looks different. He starts with the definition of n-simplex. This time the set up is not a Topological Space; it is $\mathbb{R}^n$ and first few simplexes are point, line, triangle and tetrahedron then he moves on to define a simplicial complex. Then looks at the homeomorphic images of simplicial complexes giving rise to the definition of triangulization.
So there are n-cells, simplexes, regular cell complexes. One thing I understand is that n-cells and regular cell complexes are defined for a topological space and simplexes are defined in $\mathbb{R}^n$.
Apart from my first question, what is the relation between these three?
Best Answer
This is a very common issue in topology: The same terminology (boundary and interior) have different meaning in different branches of topology. My guess is that you already took a General Topology (Point Set Topology) class and are familiar with the notion of boundary and interior of a subset $A\subset X$ in this branch of topology.
The definition of a cell you are given is a bit sloppy. A cell is a topological space $C$ homeomorphic to the closed $n$-dimensional disk $D^n$, equipped with a certain extra structure (which has a somewhat sloppy description in the document you attached). Let us ignore the extra structure. Note that the disk $D^n$ is a subset of $R^n$. The interior and the boundary of $D^n$ are defined with respect to this embedding. In other words, the interior of $D^n$ is $$ int(D^n)=\{x\in R^n: |x|< 1\} $$
and the boundary of $D^n$ is $$ \partial D^n= \{x\in R^n: |x|= 1\}= S^{n-1}. $$ What is not immediate with this definition is that the interior and the boundary of a cell are topologically invariant in the sense that a homeomorphism $h: C\to D^n$ would send interior to interior and boundary to boundary. This is a theorem (a corollary of Brouwer's Invariance of Domain theorem). Once you have this invariance, you can talk about the boundary and interior of a cell $C$, which is regarded as an abstract topological space, not embedded anywhere.
Now, a (nondegenerate) line segment $pq$ in $R^n$, $n>1$, is still a 1-cell simply because it is homeomorphic to $D^1=[-1,1]$. Of course, the interior of $pq$ in the sense as above is not the same as the interior of $pq$ as a subset of $R^n$ (the latter is empty).
If you are bothered by this "double speak", you can refer for yourself to the "interior" defined above as "cell-interior" and the boundary as the "cell-boundary". At the same time, you can call the interior of a subset $A\subset X$ in the sense of general topology as "topological interior".
Hope it helps.