a question about a concept in algebraic topology

Question

I'm reading Munkres elements of algebraic topology, and have come across a definition.

Here the book shows what a link is in some example. However, I think the link of v0 should include the lines that contain the vertex v0, not just those lines presented in the picture. Is my thought correct? Or where am I wrong? Thanks for helping

Answer 1

This is a well known typo in Munkres' book.

Answer 2

The picture for the link of $v_0$ is correct. $St(v_0)$ in this picture consists of $v_0$, the interiors of all edges meeting $v_0$, and the interiors of all triangles meeting $v_0$, i.e., kind of a polygonal "open disk". Its closure contains $v_0$, all of the (closed) edges meeting $v_0$, and all of the (closed) triangles meeting $v_0$. The difference between these two sets is exactly the edges "around the outside". The interiors of the edges meeting $v_0$ are contained in both the star and the closed star, so in the difference (the link), they are not present.