S226: Gluing of Fortissimo and Sierpinski spaces

properties/P000021.md

@@ -15,3 +15,8 @@ Every infinite set in $X$ has a limit point.
 Equivalently, every closed discrete subset of $X$ is finite.
 
 Defined on page 19 of {{zb:0386.54001}}.
+
+----
+#### Meta-properties
+
+- This property is hereditary with respect to closed sets.

spaces/S000226/README.md

@@ -0,0 +1,19 @@
+---
+uid: S000226
+name: Fortissimo space of size $\aleph_1$ and Sierpinski space glued at their limit points
+---
+
+$X$ is the quotient of the topological sum of {S155}
+and {S10} with the two limit points identified.
+In more detail, let
+- $Y=$ {S155} with $p$ as non-isolated point;
+- $Z=\{a,b\}$ be {S10} with $a$ as closed non-isolated point and $b$ as isolated point.
+
+Then $X=(Y\coprod Z)/{\sim}$ with $p$ and $a$ identified.
+
+*Note*: Both $Y$ and $Z$ embed into $X$ via the quotient map.
+For ease of notation, we'll treat them as subspaces of $X$ and
+also write $p$ for the common limit point in the quotient.
+
+$X$ could also be described as extending {S155}
+by adding one isolated point $b$ with $\overline{\{b\}}=\{b,p\}$.

spaces/S000226/properties/P000002.md

@@ -0,0 +1,7 @@
+---
+space: S000226
+property: P000002
+value: false
+---
+
+{S10} embeds into $X$ and {S10|P2}.

spaces/S000226/properties/P000018.md

@@ -0,0 +1,10 @@
+---
+space: S000226
+property: P000018
+value: true
+---
+
+Let $\mathscr U$ be an open cover of $X$.
+There is a countable $\mathscr V\subseteq\mathscr U$ with $Y\subseteq\bigcup\mathscr V$
+because {S155|P18}.
+Since every neighborhood of $p$ contains $Z$, $\mathscr V$ is also a cover of $X$.

spaces/S000226/properties/P000021.md

@@ -0,0 +1,8 @@
+---
+space: S000226
+property: P000021
+value: false
+---
+
+$Y$ is a closed subspace of $X$
+and {S155|P21}.

spaces/S000226/properties/P000114.md

@@ -0,0 +1,7 @@
+---
+space: S000226
+property: P000114
+value: true
+---
+
+By construction.

spaces/S000226/properties/P000136.md

@@ -0,0 +1,9 @@
+---
+space: S000226
+property: P000136
+value: true
+---
+
+The subspace $Y$ is closed in $X$
+and anticompact ({S155|P136}).
+So every compact subset of $X$ intersects $Y$ in a finite set, and is itself finite.

spaces/S000226/properties/P000147.md

@@ -0,0 +1,10 @@
+---
+space: S000226
+property: P000147
+value: true
+---
+
+The neighborhoods of $p$ in $X$ are the sets $V\cup Z$ with $V$ neighborhood of $p$ in $Y$.
+Since {S155|P147},
+$p$ is also a P-point in $X$.
+And every other (isolated) point is trivially a P-point.

spaces/S000226/properties/P000174.md

@@ -0,0 +1,11 @@
+---
+space: S000226
+property: P000174
+value: true
+---
+
+The neighborhoods of $p$ in $X$ are the sets $V\cup Z$ with $V$ neighborhood of $p$ in $Y$.
+Since {S155|P174},
+there is a neighborhood base $\mathscr N$ for $p$ in $Y$ that is totally ordered by inclusion.
+Then $\{V\cup Z:V\in\mathscr N\}$ is a totally ordered neighborhood base for $p$ in $X$.
+And $X$ is trivially well-based at each of the other (isolated) points.

spaces/S000226/properties/P000203.md

@@ -0,0 +1,7 @@
+---
+space: S000226
+property: P000203
+value: true
+---
+
+All points except $p$ are isolated.

spaces/S000226/properties/P000229.md

@@ -0,0 +1,12 @@
+---
+space: S000226
+property: P000229
+value: true
+---
+
+The path components of $X$ are $Z$ and the singletons $\{x\}$ with $x\in Y\setminus\{p\}$
+(because {S10|P37} and
+every $\{x\}$ with $x\in Y\setminus\{p\}$ is clopen in $X$).
+
+And since {S10|P229},
+every path component of $X$ is {P229}.

spaces/S000226/properties/P000234.md

@@ -0,0 +1,8 @@
+---
+space: S000226
+property: P000234
+value: false
+---
+
+Every neighborhood $V$ of $p$ contains a point $x\in Y\setminus\{p\}$.
+Hence $V$ is not connected since $\{x\}$ is clopen in $X$.