# Subsets $\Omega\subseteq X$ such that any continuous $f:\Omega\to Y$ can be extended continuously to $F:X\to Y$

This is a reference request. Below is a prelude to the main questions.

Let $n$ be a positive integer. Characterize all subsets $\Omega$ of $\mathbb{R}^n$ such that any continuous function $f:\Omega\to\mathbb{C}$ can be extended to a continuous function $F:\mathbb{R}^n\to\mathbb{C}$. Here, $\Omega$ inherits the subspace topology from $\mathbb{R}^n$, and of course, $\mathbb{R}^n$ and $\mathbb{C}$ are equipped with their standard topologies.

The answer to the question above is that $\Omega$ must be a closed subset of $\mathbb{R}^n$. A slightly more generalized question is given below.

Question I: What are all subsets $\Omega$ of a topological space $X$ such that any continuous function $f:\Omega\to \mathbb{C}$ can be extended continuously to $F:X\to\mathbb{C}$? As usual, $\Omega$ is equipped with the subspace topology inherited from $X$.

Of course, without any nice properties on $X$, it seems impossible to make a characterization of such subspaces $\Omega$. Hence, with some additional properties on the topology of $X$, the answer may be completely known.

I know that, if $X$ is normal, then all closed subsets $\Omega$ of $X$ have the required property, using the Tietze Extension Theorem. Is there an example of a normal topological space $X$ which contains a non-closed subset $\Omega$ such that any continuous function $f:\Omega\to\mathbb{C}$ has a continuous extension to $X$?

Question II: Even more generally, let $X$ and $Y$ be topological spaces. Find all extendable subsets $\Omega$ of $X$, namely, all subsets $\Omega\subseteq X$ such that any continuous function $f:\Omega\to Y$ can be extended continuously to $F:X\to Y$.

Of course, niceness constraints on $X$ and $Y$ may be needed for a complete characterization. For example, if $X$ is a metric space, $Y$ is a convex subset of a locally convex topological vector space, and $\Omega$ is a closed subspace of $X$, then any continuous function $f:\Omega\to Y$ can be extended to a continuous function $F:X\to Y$. See for example here (although there is no mentioning that closed sets are all such sets).

This paper discusses strongly good pairs, namely, pairs $(X,Y)$ of topological spaces such that every $\Omega\subseteq X$ is extendable. It also discusses good pairs, namely pairs $(X,Y)$ of topological spaces such that, for every $\Omega\subseteq X$ and for any continuous function $f:\Omega\to Y$, there exists a function $F:X\to Y$ extending $f$ such that $F$ is continuous at every point in $\Omega$. This gives me an idea for my final question.

Question III: Let $X$ and $Y$ be topological spaces. What are all weakly extendable subsets $\Omega$ of $X$, namely, subsets $\Omega\subseteq X$ such that, for any continuous function $f:\Omega\to Y$, there exists a function $F:X\to Y$ extending $f$ such that $F$ is continuous at every point in $\Omega$. We shall call such an extension $F$ a weakly continuous extension of $f$ (relative to $\Omega$).

In the case where $X=\mathbb{R}^n$ for some $n\in\mathbb{N}$ and $Y=\mathbb{C}$, it is known that any subset $\Omega$ of $X$ is weakly extendable. In fact, if $Y$ is a locally compact Hausdorff space and $X$ is metrizable, then $(X,Y)$ is a good pair. What are other resources where a full characterization of weakly extendable subsets is known?

• For the first question consider $\Omega = \omega_1 \subseteq X = \omega_1 + 1$, as ordinals in the order topology. – Henno Brandsma Aug 21 '16 at 5:28

Question I. A subset $\Omega$ of $X$ is extendable iff $\Omega$ is $C$-embedded in $X$. The sufficiency can be proved by the coordinatewise extension, the necessity can be proved using that $\Bbb R$ is a retract of $\Bbb C$.
Henno Brandsma’s example is quite sharp, because if $X$ is a Tychonoff space in which each point is $G_\delta$-set then each $C$-embedded subset of $X$ is closed. Indeed, for each point $x_0\in\overline{\Omega}\setminus\Omega$ it is easy to construct a continuous function $g:X\to\Bbb [0,1]$ such that $g(x)=0$ iff $x=x_0$. Then a function $f(x)=(1/g(x))|_\Omega$ cannot be extended from the set $\Omega$ to the whole space $X$.
Each closed subset of a $T_1$ space $X$ is $C$-embedded iff the space $X$ is normal.
Question II. I have the following trivial remarks. Given a pair $X$, $Y$ of topological spaces I shall call a space $X$ $Y$-collapsing, provided any continuous function from $X$ to $Y$ is constant. For instance, there exists a regular $\Bbb R$-collapsing space. Any $Y$-collapsing subset of the space $X$ is extendable. Conversely, each extendable subset of a $Y$-collapsing space $X$ is $Y$-collapsing too. In particular, if all [linearly] connected subsets of $Y$ are one-point then each [linearly] connected subset of $X$ is extendable. Conversely, if $Y$ is not connected and $X$ is connected then each extendable subset of $X$ is connected too.
If $Y$ is metrizable then any closed subset of any metrizable space $X$ is extendable iff $Y$ is an absolute extensor.
Question III. If $Y$ is an absolute neighborhood extensor then any closed subset of any metrizable space $X$ is weakly extendable.