Let \(X\) be a topological space and \(n\geq 0\). Let \(D^n\subseteq\RR^n\) be the closed unit ball and \(S^{n-1}=\del D^n\) its boundary sphere, with \(S^{-1}=\emptyset\). For a continuous map \(f\colon S^{n-1}\to X\), the space obtained by attaching an \(n\)-cell to \(X\) along \(f\) is the quotient space \[ X\cup_f D^n \coloneqq \qty{X\disjoint D^n}/\qty{s\sim f(s) \text{ for } s\in S^{n-1}}, \] the case \(X_0=X\), \(X_1=D^n\), \(A=S^{n-1}\) of attaching a space along a map. The map \(f\) is the attaching map. The composite \(\Phi\colon D^n\to X\disjoint D^n\to X\cup_f D^n\) of the inclusion and the quotient map is the characteristic map, its image \(\Phi(D^n)\) is the closed \(n\)-cell, and \(e^n\coloneqq\Phi(D^n\setminus S^{n-1})\) is the open \(n\)-cell.
In low dimensions:
\(n=0\): \(D^0\) is a point and \(S^{-1}=\emptyset\), so \(X\cup_f D^0=X\disjoint\ts{\pt}\).
\(n=1\): \(D^1=[-1,1]\subseteq\RR\) and \(f\) is a map \(S^0=\ts{-1,1}\to X\); the closed \(1\)-cell is the image of a path from \(f(-1)\) to \(f(1)\).
\(n=2\): \(D^2\subseteq\RR^2\) is the closed disk, attached along a map \(S^1\to X\).
\(n=3\): \(D^3\subseteq\RR^3\) is the closed ball, attached along a map \(S^2\to X\).