Open map theorem for metric spaces

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Semiconcave Functions in Alexandrov???s Geometry (2013, background): “’s spaces (the proof requires some quasiﬁcations, but essentially is the same), for spaces with curvature bounded above and for locally compact spaces with well deﬁned tangent cone at each point, see [Lytchak]. It was pointed out to me that some traces of these properties can be found even in genera”
The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces (2012, background): “ the closed set X¯ U.) The crucial point is that this existence result does not require any compactness of the underlying space X¯. The uniqueness result and exponential Lipschitz bound is taken from [Lyt05]. Remark 4.12. In analysis (PDEs, mathematical physics), instead of the upward gradient ﬂow m”
Gradient flows and Evolution Variational Inequalities in metric spaces. I: Structural properties (2018, unknown): “In [70] we will generalize this result to cover arbitrary Alexandrov spaces and even more general cases, characterized by a new geometric property, weaker than the Alexandrov one, based on a local angle condition between geodesics and on the semi-concavity of the squared distance [82, 35] (see also ”