A generalized Urysohn imbedding and Tychonoff fixed point theorem in topological space (Q1418410)

The authors establish the following generalized version of the Urysohn imbedding theorem: If (\(X,\tau\)) is a \(T_1\)-space, where \(\tau=\bigvee_{\alpha\in I}\tau_{\alpha}\) and, for each \(\alpha\in I\), \(\tau_{\alpha}\) is a second countable normal topology on \(X\), then \(X\) is homeomorphic to a subspace of the locally convex Hausdorff topological vector space \(E=\prod_{\alpha\in I}E_{\alpha}\), where \(E_{\alpha}=l^2\) for each \(\alpha\in I\). Let \(H:X\rightarrow E\) be the homeomorphism obtained by this theorem. Then \(X\) is a uniform space generated by the family \(\{\rho_{\alpha}^*:\alpha\in I\}\) of pseudometrics defined by \(\rho_{\alpha}^*(x,y)=p_{\alpha}(H(x)-H(y))\), for \(x,y\in X\) where, for each \(\alpha\in I\), \(p_{\alpha}\) is the seminorm in \(l^2\). A subset \(K\) of \(X\) is called \(M\)-convex if for each \(x,y\in K\) with \(x\not= y\) there exists a subset \([x,y]\) of \(K\) (called the segment joining \(x\) and \(y\)) and a homeomorphism \(h:[0,1]\rightarrow [x,y]\) such that \(h(0)=x\), \(h(1)=y\) and \(\rho_{\alpha}^*(x,h(t))=t\rho_{\alpha}^*(x,y)\), \(\rho_{\alpha}^*(y,h(t))=(1-t)\rho_{\alpha}^*(x,y)\) for all \(\alpha\in I\) and each \(t\in [0,1]\). The authors prove that if \(K\) is a compact \(M\)-convex subset of \(X\) then any continuous mapping of \(K\) into itself has a fixed point.