度量与半度量空间的弱相似性
Weak similarities of metric and semimetric spaces
摘要：
设（X，dX）和（Y，dY）是距离集分别为D（X）和D（Y）的半度量空间。映射F：X \to Y是弱相似的，如果它是满射的，并且存在严格递增的f：D（Y）\to D（X）使得dX = f \circ dY \circ F。证明了测地空间之间的弱相似是通常的相似，并且如果X和Y是超度量紧的且D（X）= D（Y），则每个弱相似F：X \to Y是等距的.给出了弱相似是同胚或一致等价的条件。
英文摘要：
Let (X,dX) and (Y,dY) be semimetric spaces with distance sets D(X) and, respectively, D(Y). A mapping F : X \to Y is a weak similarity if it is surjective and there exists a strictly increasing f : D(Y) \to D(X) such that dX = f \circ dY \circ F. It is shown that the weak similarities between geodesic spaces are usual similarities and every weak similarity F : X \to Y is an isometry if X and Y are ultrametric and compact with D(X) = D(Y). Some conditions under which the weak similarities are homeomorphisms or uniform equivalences are also found.
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