Skip to main content
added 147 characters in body
Source Link

A topologicalSpanier mentions that locally contractible implies homologically locally connected but I'm wondering whether contractible implies homologically locally connected?

Definition of homologically locally connected: "A space X$X$ is said to be homologically locally connected (based on definition in EH Spanier's "Algebraic Topology")dimension $n$ if for every $x \in X$ and neighbourhoodneighborhood $U$ of $x$ there exists a neighbourhoodneighborhood $V$ of $x$, in $V \subset U$$U$ such that $U$ and$\tilde{H}_q(V) \to \tilde{H}_q(U)$ is trivial for $V$ are homotopy equivalent$q \leq n$. Spanier mentions that locally contractible implies It is said to be homologically locally connected but I'm wondering whether contractible impliesif it is homologically locally connected? in dimension $n$ for all $n$."

A topological space X is homologically locally connected (based on definition in EH Spanier's "Algebraic Topology") if for every $x \in X$ and neighbourhood $U$ of $x$ there exists a neighbourhood $V$ of $x$, $V \subset U$ such that $U$ and $V$ are homotopy equivalent. Spanier mentions that locally contractible implies homologically locally connected but I'm wondering whether contractible implies homologically locally connected?

Spanier mentions that locally contractible implies homologically locally connected but I'm wondering whether contractible implies homologically locally connected?

Definition of homologically locally connected: "A space $X$ is said to be homologically locally connected in dimension $n$ if for every $x \in X$ and neighborhood $U$ of $x$ there exists a neighborhood $V$ of $x$ in $U$ such that $\tilde{H}_q(V) \to \tilde{H}_q(U)$ is trivial for $q \leq n$. It is said to be homologically locally connected if it is homologically locally connected in dimension $n$ for all $n$."

added 8 characters in body
Source Link

A topological space X is homologically locally connected (based on definition in EH Spanier's "Algebraic Topology") if for every $x \in X$ and neighbourhood $U$ of $x$ there exists a neighbourhood $V$ of $x$ in, $U$$V \subset U$ such that $U$ and $V$ are homotopy equivalent. Spanier mentions that locally contractible implies homologically locally connected but I'm wondering whether contractible implies homologically locally connected?

A topological space X is homologically locally connected (based on definition in EH Spanier's "Algebraic Topology") if for every $x \in X$ and neighbourhood $U$ of $x$ there exists a neighbourhood $V$ of $x$ in $U$ such that $U$ and $V$ are homotopy equivalent. Spanier mentions that locally contractible implies homologically locally connected but I'm wondering whether contractible implies homologically locally connected?

A topological space X is homologically locally connected (based on definition in EH Spanier's "Algebraic Topology") if for every $x \in X$ and neighbourhood $U$ of $x$ there exists a neighbourhood $V$ of $x$, $V \subset U$ such that $U$ and $V$ are homotopy equivalent. Spanier mentions that locally contractible implies homologically locally connected but I'm wondering whether contractible implies homologically locally connected?

added 10 characters in body
Source Link

A topological space X is homologically locally connected (based on definition in EH Spanier's "Algebraic Topology") if for every $x \in X$ and neighbourhood $U$ of $x$ there exists a neighbourhood $V$ of $x$ in $U$ such that $U$ and $V$ are homotopichomotopy equivalent. Spanier mentions that locally contractible implies homologically locally connected but I'm wondering whether contractible implies homologically locally connected?

A topological space X is homologically locally connected (based on definition in EH Spanier's "Algebraic Topology") if for every $x \in X$ and neighbourhood $U$ of $x$ there exists a neighbourhood $V$ of $x$ in $U$ such that $U$ and $V$ are homotopic. Spanier mentions that locally contractible implies homologically locally connected but I'm wondering whether contractible implies homologically locally connected?

A topological space X is homologically locally connected (based on definition in EH Spanier's "Algebraic Topology") if for every $x \in X$ and neighbourhood $U$ of $x$ there exists a neighbourhood $V$ of $x$ in $U$ such that $U$ and $V$ are homotopy equivalent. Spanier mentions that locally contractible implies homologically locally connected but I'm wondering whether contractible implies homologically locally connected?

Source Link
Loading