Consider the topological spaces $X$ with the following property: 

>For every compact $K\subseteq X$ there is a compact set $L$ such that $K\subseteq L\subseteq X$ and $L$ is a [retract][1] of $X$. 

Let us call this property RBC (Retract to a Bigger Compact). This property seems natural and useful, but I have been unable to find anything directly on it. 

It is not hard to see that the following is true: 

>**Proposition 1:** Every closed convex topological subspace $C$ of a locally convex metrizable topological vector space $V$ has the RBC property. 

*Details on Proposition 1:* Take any nonempty compact $K\subseteq C$. Then, by [known results][2] (note, in particular, Ref. [7] there), the closed convex hull $L:=\overline{\text{conv}\,K}$ of $K$ in $V$ is a retract of $V$. Since $L\subseteq C\subseteq V$, it follows that $L$ is a retract of $C$. Since $K$ is compact, $L$ is also compact, and of course $K\subseteq L\subseteq C$. Thus, $C$ has the RBC property. $\Box$


It is also clear that the RBC property is topological and thus invariant with respect to homeomorphisms. Trvially, any compact topological space has the RBC property. 

> **Question:** Can one characterize the RBC property? 

That is, can one give a usable necessary and sufficient condition for it? Or a usable sufficient condition somewhat close to necessity? By "usable", I mean without the quantifier "there is" in the definition of the RBC property. 

I do not know the answer even to this **question:** Is there an example of a topological space without the RBC property? 

Thinking about the latter question, I have in mind the "non-retract" example of the $(n-1)$-sphere, which is compact but not a retract of the corresponding closed ball, whereas the ball is compact as well and of course is a retract of itself and of the corresponding $n$-space. 

**Update:** The latter question has been answered in comments by erz, Anonymous, and Taras Banakh, who provided examples of topological spaces without the RBC property. 

At this point, I would like to make the first question, to characterize the RBC property, more specific: 

**Specific question:** Is it true that all Polish spaces have the RBC property? 

My **motivation** for all these questions comes from probability. Indeed, any retraction $r$ from $X$ to a compact subset $K$ of $X$ naturally induces the truncation map $\xi\mapsto r\circ\xi$ of random elements $\xi$ of $X$, so that the truncated version $r\circ\xi$ of $\xi$ is a random element of the compact set $K$. Moreover, this truncation map is continuous: if $\xi_t\to\xi$ in distribution, then $r\circ\xi_t\to r\circ\xi$ in distribution. 

**Update 2:** Following the latest comment by Anonymous, I have found a recent paper by [Lipham][3], where on the first page one finds an example of a connected completely metrizable space which has no compact connected subset of cardinality greater than one, and therefore does not have the RBC property. By [Mazurkiewicz's theorem][4], this space is also separable and hence Polish. Thus, it is not true that all Polish spaces have the RBC property.



  [1]: https://en.wikipedia.org/wiki/Retraction_(topology)#Retract
  [2]: https://en.wikipedia.org/wiki/Retraction_(topology)#Absolute_neighborhood_retract_(ANR)
  [3]: https://arxiv.org/pdf/1608.00292
  [4]: https://en.wikipedia.org/wiki/G%CE%B4_set#Properties