What's your favorite Cantor space?

The vertices of the Hilbert cube! This is just the countable product $[0, 1]^{\mathbb N}$ under the product topology. The Cantor set embeds as the set of vertices $\{0, 1\}^{\mathbb N}$.

Further, if one gives the Cantor set its natural metric $d(x, y) = \sum_i 2^{-i} |x_i - y_i|$, it embeds isometrically into the Hilbert cube with the same metric. Both metrics are of course compatible with their respective topologies.

$\mathcal C$ is the 1 point compactification of the upper boundary of the solvable Baumslag-Solitar group $\text{BS}(1,n)$.

When $n=p$ is prime this comes down to the statement that $\mathcal C$ is the 1 point compactification of the $p$-adic rational numbers.

The set of wild points of the Alexander horned sphere is a (tame) Cantor set.

The Alexander horned sphere $S_A$ is an example of a $2$-sphere in $\mathbb{R}^3$ that is not ambiently homeomorphic to the standard $2$-sphere. We say a $2$-sphere $S$ is locally flat at a point $x\in S$ if there exists an open neighbourhood $U$ of $x$ in $\mathbb{R}^3$ such that $(U, U\cap S)\cong (\mathbb{R}^3, \mathbb{R}^2)$ and the points at which $S$ fails to be locally flat are called wild points. It turns out the set of ''ends'' of horns of the Alexander horned sphere with induced metric is a Cantor set in $\mathbb{R}^3$ that is also ambiently homeomorphic to the standard ternary Cantor set.

In probability, the Cantor distribution has a continuous Cumulative Distribution Function, even though its support, the Cantor set $\mathcal C$, is nowhere dense. So the distribution has no points of positive probability and no positive density anywhere.

The set of zeros of Brownian motion $W:[0,1]\to\mathbb R$ is almost surely a Cantor set (of Hausdorff dimension 1/2). So is the set of zeros of a generic continuous $f:[0,1]\to\mathbb R$, provided it is not empty (this one has dimension 0).

Edit: The topological version is also true in higher dimensions: for continuous functions $f:[0,1]^d\to\mathbb R^d$, the zero set is generically a Cantor set of Hausdorff dimension 0.

The universal cover of a connect-sum of $3$-dimensional lens spaces is generally the complement of a Cantor set in $S^3$. If you give the lens spaces their spherical structures the Cantor set is essentially uniquely defined, up to a conformal transformation of $S^3$.

Generic compact subset of $[0,1]$ or of $[0,1]^2$ or of any non-empty Polish space without isolated points.

Another one: the hyperspace of compact subsets of the Cantor set.

Every uncountable Polish space contains a homeomorphic copy of the Cantor space.

In the descriptive set theory course I'm following, this was used last week to prove the continuum hypothesis for Borel sets, i.e., a Borel subset of a Polish space (e.g., $ℝ$) is at most countable or has cardinality continuum. The other ingredient is the fact that given a Borel subset of a Polish space, one can refine the topology to a new Polish topology, without changing the Borel subsets, so that this Borel subset becomes clopen (hence $G_δ$, hence Polish with the subspace topology).

The set of Penrose tilings $\Omega_0$ with a vertex at the origin forms a Cantor set with the usual tiling metric (tilings are close if they agree, up to a small translation, on a large radius ball around the origin).

This comes from the very useful result that the tiling space $\Omega$ of any repetitive, aperiodic tiling with finite local complexity is a Fiber bundle over the torus with Cantor set fiber (arxiv link).

You can think of it as coming from the fact that any finite patch in the tiling repeats infinitely often (repetitivity), and so to make a 'small jump' in $\Omega_0$, just translate your current tiling to one that looks the same out to a large radius. Because of aperiodicity, the two tilings are different, so it's a non-zero distance away in the tiling metric. Then, connected components in $\Omega_0$ are singletons because we're not including any tilings where the origin is not a vertex, so we have to make discrete 'jumps'.

I have long had a favorite Cantor space: $\prod \mathbb F_2$, the countable direct product of the the field of two elements. First of all, I'm charmed by its being a topological group — with its unique invariant probability measure. And of course it's a topological vector space.

Let $\mathbf U(\mathbb H)$ denote the unitary group of the separable infinite-dimensional real Hilbert space $\mathbb H$. Think of $\mathbf U(\mathbb H)$ as acting on the unit sphere $\mathbb S \subset \mathbb H$.

My favorite instance of this group $\prod \mathbb F_2$ is the closed subgroup of $\mathbf U(\mathbb H)$ consisting of all elements that negate some subset of coordinates.

This is pretty cool: A compact topological vector space with an invariant probability measure acting by isometries on the unit sphere of the unique complete real separable infinite-dimensional topological vector space.

It's hard to pick a favorite, but one I have come across lately is a result of Glasner and Weiss. The symmetric group on $\mathbb{Z}$ (when given the topology of pointwise convergence) has the Cantor set as it's universal minimal flow.

The path space of a Bratelli diagram (with minor restriction so there are no atoms) is homeomorphic to a Cantor set. Vershik used this to describe dynamical systems measure-theoretically (a single automorphism of a measure space), and this was extended by Putnam et al to describe the topological version (that is, every $(X,\alpha)$ where $X$ is a Cantor space, and $\alpha$ is a self-homeomorphism and minimal), arises (up to the obvious notion of equivalence) from some Bratelli diagram of a simple dimension group together with a total ordering on the path space.

My favorite version of the Cantor space is the set of all subsets of $\mathbb{N}$, imbued with the topology where the neighborhood basis for a set $S\subseteq \mathbb{N}$ is the collection $U_n$ of sets that are disjoint from $S$ over the first $n$ numbers: $$U_n := \{T\subseteq \mathbb{N}\ |\ (T\triangle S)\cap \{0,1,\ldots,n\} =\varnothing\}$$ for $n\in \mathbb{N}$.