Proposition 4.3.4.11. Let $X$ and $Y$ be simplicial sets. If either $X$ or $Y$ is finite, then the join comparison map $T_{X,Y}: |X \star Y| \rightarrow |X| \star |Y|$ of Construction 4.3.4.9 is a homeomorphism.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Without loss of generality, we may assume that $X$ is finite. Then the geometric realization $|X|$ is a compact Hausdorff space (Corollary 3.6.1.12). Using Remarks 4.3.4.6 and 4.3.3.31, we see that the functors
\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Top}_{|X|/} \quad \quad Y \mapsto |X \star Y|, Y \mapsto |X| \star |Y| \]
preserve colimits. Consequently, if we regard $X$ as fixed, then the collection of simplicial sets $Y$ for which $T_{X,Y}$ is a homeomorphism is closed under colimits. Since every simplicial set can be realized as a colimit of standard simplices (Remark 1.1.3.13), it will suffice to prove Proposition 4.3.4.11 in the special case where $Y = \Delta ^{q}$ is a standard simplex. In this case, $Y$ is also finite. Repeating the preceding argument (with the roles of $X$ and $Y$ reversed), we are reduced to proving that $T_{X,Y}$ is a homeomorphism in the case where $X = \Delta ^{p}$ is also a standard simplex. In this case, the desired result follows from Example 4.3.4.10. $\square$