Confused about the definition of 1-cell There are two questions. 2nd question is placed below the image file.

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*I am confused about the definition of 1-cell. My textbook says that for $n > 0$, $n$-cell is obtained by attaching $D^n$ to space $X$ with a map $f: S^{n-1} \rightarrow X$. This makes sense when $n \neq 1$. However, I am confused when $n = 1$. Take $X = {x_0}$ and attach 1-cell to $X$. $S^0 = \{a, b\}$, so identify two points on $S^1$. The resulting space looks like $S^1 \vee S^1$. Is this the correct interpretation?


*How would I describe a straight line segment (with two endpoints included) in the language of cell-complex?


You must attached $D^n$, and not $S^n$, via a map $S^{n-1}\to X$. $D^1$ is a segment, so such a map attaches a segment by fixing the two endpoints. This should also answer your second question.
