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.
