diff --git a/spaces/S000131/properties/P000183.md b/spaces/S000131/properties/P000183.md
index 7e59be3ad..8b42af08d 100644
--- a/spaces/S000131/properties/P000183.md
+++ b/spaces/S000131/properties/P000183.md
@@ -7,7 +7,7 @@ refs:
     name: Answer to "Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?"
 ---
 
-Each spine $C_m=(\{m\}\times\omega)\cup\{\infty\}$ is a closed subspace of $X$ homeomorphic to a convergent sequence ({S20});
+Each spine $C_m=(\{m\}\times\omega)\cup\{\infty\}$ is a closed subspace of $X$ homeomorphic to a convergent sequence (in the {S20});
 and {S20|P183}.
 And every compact subset of $X$ is contained in the union of a finite number of these spines.
 
diff --git a/spaces/S000139/properties/P000064.md b/spaces/S000139/properties/P000064.md
index 801f3b3ad..7be7e5153 100644
--- a/spaces/S000139/properties/P000064.md
+++ b/spaces/S000139/properties/P000064.md
@@ -7,4 +7,4 @@ refs:
     name: Baire space on Wikipedia
 ---
 
-The subspace $X\setminus\{\infty\}$ is Baire (because locally compact Hausdorff) and dense in $X$.
+The subspace $X\setminus\{\infty\}$ is Baire (because it is locally compact and Hausdorff) and dense in $X$.
diff --git a/spaces/S000139/properties/P000183.md b/spaces/S000139/properties/P000183.md
index b97344c4a..e9e25a197 100644
--- a/spaces/S000139/properties/P000183.md
+++ b/spaces/S000139/properties/P000183.md
@@ -7,4 +7,4 @@ value: true
 Each of the circles (corresponding to an interval $[n,n+1]$, $n\in\mathbb Z$, with the endpoints identified) is a closed subspace of $X$ and {S170|P183}.
 And every compact subset of $X$ is contained in the union of a finite number of these circles.
 
-Therefore, taking a countable $k$-network from each of the (countably many) circles and forming their union gives a countable $k$-network for $X.$
+Therefore, taking a countable $k$-network from each of the (countably many) circles and forming their union gives a countable $k$-network for $X$.
diff --git a/spaces/S000170/README.md b/spaces/S000170/README.md
index 44fa7568d..413a623ff 100644
--- a/spaces/S000170/README.md
+++ b/spaces/S000170/README.md
@@ -2,7 +2,7 @@
 uid: S000170
 name: Circle
 aliases:
-- S1
+- $S_1$
 - One-dimensional sphere
 refs:
   - wikipedia: Circle