Existence of a compact Hausdorff space which continuously embeds into itself as a set with no interior.

Question

Is there an example of a compact Hausdorff space $X$ together with a continuous injection $\iota \colon X \hookrightarrow X$ such that $\iota(X)$ has empty interior (relative to $X$)?

Answer 1

Let $\, Z\ne\emptyset\, $ be an arbitrary Hausdorff compact space such that $\, Z\, $ has no isolated point (is dense in itself) and $\, X\, := Z^2\, $ is homeomorphic to $\, Z.\, $ The Hilbert cube and Cantor discontinuum are such $\, Z.\, $

Also, for every non-empty Hausdorff compact space $\, Y\, $ which has no isolated point, let $\, Z\, $ be a Tikhonov infinite product of copies of $\, Y.\, $ Then again $\, X:=Z^2\, $ is homeomorphic to $\, Z.\, $

In all these examples of $\, X := Z^2,\, $ it is homeomorphic to $\, Z\times\{p\}\, $ where $\, p\in X\, $ is arbitrary. And $\, Z\times\{p\}\, $ has empty interior in $\, X.\, $ Thus, each such space $\, X\, $ provides an example for the  OP's Question.

Answer 2

Here is an example, based on the Cantor discontinuum $\, C\, $ which consists of all real numbers $\, r\in[0;1]\, $ which have only $\, 0\, $ and $\, 2\, $ in their ternary positional decomposition.

Let $\, X\, :=\, C^2\, \subseteq\, \mathbb R^2,\, $ and let $\, h:X\rightarrow C\times\{0\}\, $ be a homeorphism (it's well known that it exists). Then $\, C\times\{0\}\, $ has empty interior in $\, X.\, $ To be pedantic, consider the homeomorphic embedding of $\, X\, $ into $\, X\, $ (i.e. injective continuus map) induced by $\, h,\, $ and we are done.   Great!

EXTRA:

Starting from an arbitrary requested example $\,(X\ A)\,$ (where $\, X\ A\,$ are homeomorphic, and $\,A\subseteq X\, $ has empty interior with respect to $\,X),\,$ we obtain further examples of this type:

$$ (X\times Q\quad A\times Q) $$

where $Q$ is an arbitrary non-empty (compact) topological space.

This way, if we start with a $0$-dimensional $X,\,$ we can obtain examples of arbitrary finite or infinite dimensions.