Originally Posted by

**J Graham** A topological space X is said to be locally path-connected (or locally arcwise connected) if, for any point x, every open neighborhood U of x contains an OPEN path-connected set V containing x. This is essentially the definition given in Massey, A Basic Course in Algebraic Topology, p. 36, and other books use an equivalent definition (for example, this is exactly Corollary 1.19, p. 28, in Rotman, An Introduction to Algebraic Topology).

The definition in Massey actually says, " . . . if each point has a basic family of arcwise connected neighborhoods," but I assume the word "neighborhood" here means an OPEN set containing the point, as it usually does, and the above expresses this, I believe.

Later on, however, in Exercise 2.1 on p. 123, Massey asks the reader to prove that the definition above is equivalent to the definition WITH THE WORD "OPEN" OMITTED. That is, the condition is that U above must contain a path-connected set V containing x (NOT nec. an open one). This makes no sense to me, because such a set could then be {x} itself, and thus any space would be locally path connected.

I am obviously missing something here. Can anyone enlighten me?

Thanks.