Discrete spaces are CW complexes

properties/P000240.md

@@ -22,8 +22,8 @@ The $0$-skeleton $X_0$ has the discrete topology; if it is empty, $X$ itself is
 
 Here $D^n$ is the closed unit disk in $\mathbb R^n$ and $\partial D^n=S^{n-1}$ is the unit sphere in $\mathbb R^n$. We set $\partial D^0=S^{-1}=\emptyset$ by convention.
 
-*Note*: A *CW-structure* on a topological space $X$ is a filtration $X_{-1}\subseteq X_0\subseteq X_1 \subseteq\dots$ satisfying the conditions above.
-Strictly speaking, a *CW complex* is a space $X$ together with a compatible CW-structure.
+*Note*: A *CW structure* on a topological space $X$ is a filtration $\emptyset = X_{-1}\subseteq X_0\subseteq X_1 \subseteq\dots$ satisfying the conditions above.
+Strictly speaking, a *CW complex* is a space $X$ together with a compatible CW structure.
 For simplicity and by a slight abuse of language, we call the space itself a CW complex if it admits a CW structure.
 
 Defined on page 5 of {{zb:1044.55001}}, also given in Theorem II.2.4 of {{zb:0207.21704}}.

theorems/T000889.md

@@ -0,0 +1,9 @@
+---
+uid: T000889
+if:
+  P000052: true
+then:
+  P000240: true
+---
+
+Choose $X_0=X$.