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In recent years, cosmic shear has emerged as a powerful tool to study the statistical distribution of matter in our Universe. Apart from the standard two-point correlation functions, several alternative methods like peak count statistics offer competitive results. Here we show that persistent homology, a tool from topological data analysis, can extract more cosmological information than previous methods from the same dataset. For this, we use persistent Betti numbers to efficiently summarise the full topological structure of weak lensing aperture mass maps. This method can be seen as an extension of the peak count statistics, in which we additionally capture information about the environment surrounding the maxima. We first demonstrate the performance in a mock analysis of the KiDS+VIKING-450 data: we extract the Betti functions from a suite of $w$CDM $N$-body simulations and use these to train a Gaussian process emulator that provides rapid model predictions; we next run a Markov-Chain Monte Carlo analysis on independent mock data to infer the cosmological parameters and their uncertainty. When comparing our results, we recover the input cosmology and achieve a constraining power on $S_8 \equiv σ_8\sqrt{Ω_\mathrm{m}/0.3}$ that is 5% tighter than that of peak count statistics. Performing the same analysis on 100 deg$^2$ of Euclid-like simulations, we are able to improve the constraints on $S_8$ and $Ω_\mathrm{m}$ by 18% and 10%, respectively, while breaking some of the degeneracy between $S_8$ and the dark energy equation of state. To our knowledge, the methods presented here are the most powerful topological tools to constrain cosmological parameters with lensing data.