For questions about CW complexes (topological spaces which are built up using balls of varying dimensions known as cells).

Let $B^n$ denote the $n$-dimensional closed ball.

If $X$ is a topological space and $\varphi : \partial B^n \to X$ is a continuous map, the *adjunction space* is $X\cup_{\varphi} B^n := (X\coprod B^n)/\sim$ where $\sim$ identifies $x$ with $f(x)$. The process of going from $X$ to $X\cup_{\varphi} B^n$ is often referred to as attaching an $n$-cell and the map $\varphi$ is called the *attaching map*.

Let $X_0$ be a discrete space. Let $X_n$ be a space which can be obtained from $X_{n-1}$ by attaching $n$-cells. Then the space $X = \bigcup_{n=0}^{\infty}X_n$, topologised appropriately, is called a *CW complex*, and the spaces $\bigcup_{n=0}^kX_n$ are called the $k$-*skeletons* of $X$.

Surprisingly, not every topological space is a CW complex. For example, the Hawaiian earring does not even have the homotopy type of a CW complex.