continuous extension and smooth extension of a function

Question

Let $X$ be a metric space. Let $E$ be a subset of $X$.

(1). any continuous function $f:E\longrightarrow \mathbb{R}$ can be extended to a continuous function $g: X\longrightarrow \mathbb{R}$ such that $g|_E=f$.

(2). $E$ is closed in $X$.

are (1), (2) equivalent?

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Let $E$ be a subset of $\mathbb{R}^n$.

(3). any smooth function $f:E\longrightarrow \mathbb{R}$ can be extended to a smooth function $g: \mathbb{R}^n\longrightarrow \mathbb{R}$ such that $g|_E=f$.

(4). $E$ is closed in $\mathbb{R}^n$.

are (3), (4) equivalent?

Answer 1

$(1)\implies (2)$. Suppose $E$ is not closed. Let $a \in \overline{E}\setminus E$. The function $f(x) = 1/d(x,a)$ is continuous on $E$, but does not have a continuous extension to $X$ (it's not locally bounded at $a$).

$(2)\implies (1)$ is a classical theorem, Tietze extension.

$(3)\implies (4)$ is just like $(1)\implies (2)$.

$(4)\implies (3)$ is undefined until you specify what a smooth function on a closed set means. Under one definition, it is a function that has a smooth extension to a neighborhood of the set; then the implication is true. For another version of this implication, see Whitney extension.