Definitions and basic theorems

Author: GeoffreySangston

properties/P000224.md

@@ -1,11 +1,6 @@
 ---
 uid: P000224
 name: Weakly locally contractible
-refs:
-  - zb: "0087.38203"
-    name: On fiber spaces (Fadell)
-  - zb: "0642.54014"
-    name: LECS, local mixers, topological groups and special products (Borges)
 ---
 
 Every point of $X$ has a neighborhood which is {P199}. 
@@ -14,4 +9,4 @@ The name we have chosen for this property conforms to the [pi-base naming conven
 However, we have not seen this property mentioned with a specific name in the literature.
 
 The terminology "weakly locally contractible" has been used for multiple concepts different from this one, but the name is not standardized.
-A relatively common usage among those is as a synonym for "semilocally contractible" (see {{zb:0087.38203}} and {{zb:0642.54014}}).
+A relatively common usage among those is as a synonym for {P239}.

properties/P000239.md

@@ -0,0 +1,33 @@
+---
+uid: P000239
+name: Semilocally contractible
+aliases:
+  - Weakly locally contractible
+  - wlc
+refs:
+  - zb: "0087.38203"
+    name: On fiber spaces (Fadell)
+  - zb: "0506.54015"
+    name: A characterization of local equi-connectedness (Sakai)
+---
+
+Every point $x \in X$ has a neighborhood (equivalently, an open neighborhood) $U$ that is
+contractible in $X$; i.e., such that the inclusion map
+$U \hookrightarrow X$ is a null-homotopy.
+
+Equivalently, every point $x \in X$ has a neighborhood
+(equivalently, an open neighborhood) $U$ of
+$x$ such that the inclusion map $U \hookrightarrow X$ is
+homotopic to the constant map with value $x$.
+
+Defined as "weakly locally contractible" and given the
+acronym "wlc" in {{zb:0087.38203}}. Defined as "semi-locally contractible" in {{zb:0506.54015}}.
+
+----
+#### Meta-properties
+
+- This property is hereditary with respect to clopen sets.
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
+- This property is preserved by arbitrary disjoint unions.
+- This property is preserved by finite products.
+- If each point has a neighborhood with the property, $X$ also has the property.

theorems/T000873.md

@@ -0,0 +1,9 @@
+---
+uid: T000873
+if:
+  P000239: true
+then:
+  P000229: true
+---
+
+Immediate from the definitions.

theorems/T000874.md

@@ -0,0 +1,9 @@
+---
+uid: T000874
+if:
+  P000224: true
+then:
+  P000239: true
+---
+
+Immediate from the definitions.

theorems/T000875.md

@@ -0,0 +1,10 @@
+---
+uid: T000875
+if:
+  P000225: true
+then:
+  P000239: true
+---
+
+For subsets $V \subset U \subset X$, if $V$
+is contractible in $U$, then it is contractible in $X$.