On (co)homology locally connected spaces (Q1612185)

A topological space \(X\) is homology (cohomology, respectively) locally connected if for each point \(x \in X\) and neighborhood \(N\) of \(x\), there exists a neighborhood \(M \subset N\) of \(x\) such that the homomorphism induced by the inclusion \(H_{\ast}^s(M,\{x\}) \to H_{\ast}^s(N,\{x\})\) (\(\check{H}^{\ast}(N,\{x\}) \to \check{H}^{\ast}(M,\{x\})\), respectively) is trivial, and we call the space an \textit{HLC}-space (\textit{clc}-space, respectively). Here \(H_{\ast}^s(Y)\) and \(\check{H}^{\ast}(Y)\) are singular homology and Čech cohomology groups of a topological space \(Y\) with integer coefficients. The authors show that there exists a two-dimensional compact metric space \(X\) such that (1) \(X\) is acyclic in Čech cohomology; (2) \(X\) is a \textit{clc}-space, and (3) \(X\) is not an \textit{HLC}-space. The result is an extension of \textit{G. E. Bredon's} example in [Sheaf Theory (GTM vol.170, 131--132, Springer-Verlag) (1997; Zbl 0874.55001)], and an alternative construction by \textit{H. G. Griffith} [Proc. Lond. Math. Soc., III. Ser. 6, 455--480 (1956; Zbl 0071.01902)].