The geometric realization of an abstract simplicial complex

Let $(V,K)$ be an abstract simplicial complex. Its geometric realization $\mathrm{\mid }K\mathrm{\mid }$ is the set of functions $\alpha :V\to [0,1]$ such that:

1. $\alpha (v)=0$ for all but finitely many $v\in V$;
2. ${\sum }_{v\in V}\alpha (v)=1$;
3. the support $\mathrm{supp}(\alpha ):=\{v\in V:\alpha (v)\mathrm{\ne }0\}$ is a simplex of $K$.

For each simplex $\sigma =\{v_0,\dots ,v_n\}$ of $K$, write $\mathrm{\mid }\sigma \mathrm{\mid }:=\{\alpha \in \mathrm{\mid }K\mathrm{\mid }:\mathrm{supp}(\alpha )\subseteq \sigma \}\mathrm{.}$ Sending $\alpha$ to the barycentric tuple $(\alpha (v_0),\dots ,\alpha (v_n))$ identifies $\mathrm{\mid }\sigma \mathrm{\mid }$ with the geometric simplex spanned by the standard basis vectors indexed by $v_0,\dots ,v_n$, so $\mathrm{\mid }\sigma \mathrm{\mid }$ carries its Euclidean simplex topology.

We give $\mathrm{\mid }K\mathrm{\mid }$ the weak topology with respect to these simplex inclusions: a subset $U\subseteq \mathrm{\mid }K\mathrm{\mid }$ is declared open exactly when $U\cap \mathrm{\mid }\sigma \mathrm{\mid }$ is open in $\mathrm{\mid }\sigma \mathrm{\mid }$ for every simplex $\sigma$ of $K$.