"Open" Cell? (Hatcher and Husemoller)

Question

Have $\mathbb D^n$ be some n-dimensional unit disk. Have $\partial S$ denote the boundary of some space $S$. Let infix $(-)$ denote a difference between collections of objects.

In Dale Husemoller's Fibre Bundles (and Hatcher's Algebraic Topology) an "open" cell (and perhaps by implication a "closed" cell) is mentioned without explanation (I searched through it in Hatcher and he mentioned it once in his appendix without clarifying). We can denote by $$\mathbf e^n = \mathbb D^n - \partial \, \mathbb D^n$$ an n-cell. However, what would make this cell "open" or "closed"? I would guess that there's some topological space $(x, X)$ where $$(x - \mathbf e^n) \in X$$ implies that $\mathbf e^n$ is closed, and likewise

$$\mathbf e^n \in X$$

implies that $\mathbf e^n$ is open. This is only a guess, and even if true I would not know what base set $x$ and topology $X$ would be choice.

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I would be grateful if any answers that involve topological spaces use the full $(\star , \bigstar)$ notation and avoid any ambiguity when referring to the underlying set $(\star)$ and the topology $\big( \bigstar \big)$ placed on it. Thank you in advance for your help.

Answer 1

Hatcher defines in Standard Notations on p.xii

$e^n$: an $n$-cell, homeomorphic to the open $n$-disk $D^n-\partial D^n$.

The phrase "open $n$-cell" is synonymous with "$n$-cell", but typically avoided as it can be misleading (an open $n$-cell in a topological space need not be an open subset thereof). The one instance of this phrase in Hatcher's book is presumably just a slip of the tongue. The one appearance in Husemoller's book is in the introduction, where he does not define the terminology, but redirects the reader to J.H.C. Whitehead's paper Combinatorial Homotopy. I. There, at the start of Section $4$, we find

By a cell complex, $K$, or simply a complex, we mean a Hausdorff space, which is the union of disjoint (open) cells, to be denoted by $e$, $e^n$, $e_i^n$, etc., subject to the following condition.

This informs us that "open cell" and "cell" are the same concept and the adjective "open" is considered negligible.

The term "closed cell" is ambiguous. It does not seem to appear in Husemoller's book. In Hatcher's book, there is one instance on p.535. He calls a CW-complex regular if the characteristic maps can be chosen to be embeddings. In this case, the closures of the $n$-cells of this CW-complex are homeomorphic to the (closed) disk $D^n$ and he calls them "closed cells". However, in other contexts, one might want to refer to the closure of any cell in any CW-complex (which is not always homeomorphic to a closed disk) as a "closed cell" (this is done e.g. in Bredon's Topology and Geometry).

Answer 2

This is just an extended comment to Thorgott's answer.

There is no standard interpretation of the phrase "cell of a CW-complex $X$".

Hatcher's "$n$-cells" are homeomorphic to open $n$-disks, but they need not be open subsets of $X$.

Switzer [Switzer, Robert M. Algebraic topology-homotopy and homology. Springer, 2017] uses the word "$n$-cell" for what Hatcher would call the closure of an open cell, i.e. for the image of a characteristic map $\phi : D^n \to X$. These are in general no cells in the sense that they are homeomorphic to $D^n$. As the interior of an $n$-cell $e^n$ he denotes $\mathring e^n = e^n \setminus X^{n-1}$ - again misleading because it is general not the topological interior of $e^n$ in $X$. However, Switzer's cell-interiors are nothing else than Hatcher's cells.

Fritsch and Piccinini [Fritsch, Rudolf, and Renzo Piccinini. Cellular structures in topology. Vol. 19. Cambridge University Press, 1990] define a subspace $e$ of a space X to be

• an open $n$-cell in $X$, if it is an open $n$-ball (recall that an open $n$-ball is a space homeomorphic to the open ball $B^n$);
• a closed $n$-cell in $X$, if it is the closure (in $X$) of an open $n$-cell;
• a regular $n$-cell in $X$, if it is an open $n$-cell whose closure is an $n$-ball and whose boundary in the closure is an $(n-1)$-sphere;
• a closed regular $n$-cell in $X$, if it is the closure of a regular $n$-cell.