Contractible vs. Contractible in a space

Question

I am reading Introduction to Homotopy Theory by Arkowitz Martin and on page 9 it reads:

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More generally, if $A$ is a subset of $X$ with inclusion map $i : A \to X$; then $A$ is contractible in $X$ if $i$ is nullhomotopic.

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So if $S^1$ being the unit circle naturally embedded into $\mathbb{R}^2$, then $S^1$ is contractible in $\mathbb{R}^2$? (There is a linear homotopy from the natural embedding to the constant map to the orgin of $\mathbb{R}^2$)

Obviously $S^1$ itself is not contractible. Then intuitively what does "$S^1$ is contractible in $\mathbb{R}^2$" mean?

Answer 1

As Bongers said, any subset of $\mathbb{R}^n$ is contractible.

"$S^1$ is contractible in $\mathbb{R}^2$" just means that the circle $S^1$ indeed can be contracted to the origin in the plane as you said.