I'm a lil bit confused with the example 3 from Munkres, that the topology of $|K|$ is larger than the topology $|K|$ inherits as a subspace of $\mathbb{R}^n$.
Let $K$ be the collection of 1-simplices $\sigma_1,\sigma_2,...$ and their vertices, where $\sigma_i$ is the 1-simplex in $\mathbb{R}^2$ having vertices $0$ and $(1,1/i)$. Then K is a simplicial complex. The intersection of the underlying space $|K|$ with the open parabolic arc $\{(x,x^2):x>0\}$ is closed in $|K|$, because its intersection with each simplex $\sigma_i$ is a single point (This is clear). But why it is not closed in $\mathbb{R}^2$? The intersection is the set of points whose it's limit point is $0$, and $0$ is in the set, so it should also be closed in $\mathbb{R}^2$, no?
The point is that it's not the intersection of any closed set with the subspace, so it's not in the subspace topology. This is because any such closed set would contain $0$.