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In this note we describe closed subsets of the real line $P\subset {\mathbb R}$ for which there exists a continuous function from $P$ onto $P^2$, called a Peano function. Our characterization of those sets is based on the number of connected components of $P$. We also include a few remarks on compact subsets of $\mathbb{R}^2$ admitting Peano functions, expressed in terms of connectedness and local connectedness.

In this paper we investigate for which closed subsets \(P\) of the real line \(\mathbb{R}\) there exists a continuous map from \(P\) onto \(P^2\) and, if such a function exists, how smooth can it be. We show that there exists an infinitely many times differentiable function \(f\colon\mathbb{R}\to\mathbb{R}^2\) which maps an unbounded perfect set \(P\) onto \(P^2\). At the same time, no continuously differentiable function \(f\colon\mathbb{R}\to\mathbb{R}^2\) can map a compact perfect set onto its square. Finally, we show that a disconnected compact perfect set \(P\) admits a continuous function from \(P\) onto \(P^2\) if, and only if, \(P\) has uncountably many connected components.

Strain is a commonly used tool to tune the optoelectronic properties of semiconductors. It is especially effective for transition metal dichalcogenides (TMDs), which can withstand extreme strain up to 10%. To date, localised strain fields have been applied by transferring TMDs flakes onto a patterned SiO2 substrate. Here, we present a novel approach, where the strain in MoS2 monolayer is induced by an array of homoepitaxially grown GaAs nanomembranes. This represents a first step towards the integration of TMD monolayers with III-V semiconductor nanostructures, which is essential to develop scalable nanophotonic platforms. The strain imposed by the nanomembrane lifts the degeneracy of the exciton states, leading to linearly polarised emission. The principal axis of the linear polarisation of the emission is strictly determined by the orientation of the nanomembranes. This result is fully consistent with the expected broken crystal symmetry resulting from the imposed uniaxial strain.

The integration of transition-metal dichalcogenides (TMDs) with non-planar substrates such as nanopillars provides a way to spatially modify the optical properties mainly through the localized strain. Similar studies to date have utilized insulating SiO2 nanopillars. Here, we combine monolayer MoS2 with free standing GaAs nanowires (NWs), in views of coupling their semiconducting properties. We find that monolayer MoS2 exhibits three different configurations: pierced, wrapped and tent-like. We demonstrate how to identify the configurations by optical microscopy and elucidate the impact on the vibrational and luminescence characteristics by confocal spectroscopy mapping. In particular, we highlight the increase of intensity and shift due to the photonic properties of nanowires and increase in dielectric screening associated with the GaAs NW. This work signifies the first step towards the use of vertical III-V NW arrays as a versatile platform for spatially engineering the optical properties of TMDs.