Poset associated with a regular cell complex

Question

I am studying regular cell complexes following Justin Curry's thesis ''Sheaves, cosheaves and applications'' and I cannot prove that the set of indices $P_X$ of the definition of regular cell complex (Definition 4.1.1) is a poset with the given order ($\sigma\leq \tau$ if $X_\sigma\subseteq \overline{X_\tau}$), specifically I cannot see the antisymmetric property: if $\sigma\leq \tau$ and $\tau\leq \sigma$ then $\tau$ and $\sigma$ are equal, that is, if $X_\sigma\subseteq \overline{X_\tau}$ and $X_\tau\subseteq \overline{X_\sigma}$ then $X_\sigma=X_\tau$.

Answer 1

The two key parts of the definition for showing this are:

1. $\{X_\sigma\}_{\sigma \in P_X}$ partitions $X$
2. If $X_\sigma \cap {\bar X}_\tau \not= \emptyset$ then $X_\sigma \subseteq \bar{X}_\tau$

First we note that if $X_\sigma \subset \bar{X}_\tau$ that $\bar{X}_\sigma \subset \bar{X}_\tau$ as the closure of a set $X$ is the smallest closed set containing $X$.

Then, as $X_\sigma \subset \bar{X}_\tau$ and $X_\tau \subset \bar{X}_\sigma$ we have $\bar{X}_\sigma = \bar{X}_\tau$.

Finally as $\{X_\sigma\}_{\sigma \in P_X}$ is a partition of $X$ it is the case that $X_\sigma \cap X_\tau = \emptyset$ for $\tau \neq \sigma$. Thus if $\bar{X}_\sigma = \bar{X}_\tau$ it must be the case that $\sigma = \tau$