---- > [!definition] Definition. ([[countable complement topology]]) > Let $X$ be any set; put $\tau_{c}:\{ U \subset X : X \cut U \text{ is countable or is all of } X \}$. > \ > $\tau _c$ is a [[topological space|topology]] on $X$, called the **countable complement topology**. ^b3e080 > [!justification] > We will employ [[De Morgan's Laws]]. We need to show $\tau_{c}$ is indeed a [[topological space|topology]] on $X$: >1. $X-\emptyset = X \in \tau_{c}$ by assumption, and $X-X=\emptyset$ is finite and thus in $\tau_{c}$. >2. Let $\{ U_{\alpha} \}$ be an indexed family of elements in $\tau_{c}$. Then $X-\bigcup_{\alpha}^{} U_{\alpha}=\bigcap_{\alpha}^{}(X-U_{\alpha})$ is countable (or $X$ if the $U_{\alpha}$ are empty) as an intersection of countable sets, so $\bigcup_{\alpha}^{} U_{\alpha} \in \tau_{c}$. >3. Let $U_{1},\dots, U_{n}$ be a collection of elements in $\tau_{c}$. Then $X-\bigcap_{i=1}^{n} U_{i}=\bigcup_{i=1}^{n}(X-U_{i})$ and this set is countable (or $X$ if the $U_{i}$ are empty) as a finite union of countable sets. ^2c6df8 > [!basicnonexample] Warning. > The collection $\tau_{\infty}:= \{ U : X \cut U \text{ is infinite or empty or all of } X \}$ > is *not* a [[topological space|topology]] on $X$. As a counterexample, let $X=\mathbb{R}$, $U_{1}:=(-\infty, 0)$, $U_{2}=(0,\infty)$. Then $U_{1}$ and $U_{2}$ are in $\tau_{\infty}$, but $U_{1} \cup U_{2}$ is not. ^3208ed > [!basicproperties] Property. (Converges in countable complement $\iff$ converges in discrete $\iff$ eventually constant) > Let $\tau_{1}:= \mathcal{P}(\mathbb{R})$ be the [[discrete topology]] on $\mathbb{R}$ and $\tau_{2}$ be the [[countable complement topology]] on $\mathbb{R}$. We want to show that for any [[sequence]] $(x_{n})_{n \in \mathbb{N}} \subset \mathbb{R}$, the [[sequence]] [[converge|converges]] to $x$ for $\tau_{1}$ if and only if it [[converge|converges]] to $x$ for $\tau_{2}$. > **Lemma. $\{ x \} \cup (\mathbb{R} - \{ x_{n}: n \geq 0 \})$ is open wrt $\tau_{2}$,** since $\begin{align} \mathbb{R} - (\{ x \} \cup \mathbb{R} - \{ x_{n} : n \in \mathbb{N} \})= & ( \mathbb{R} - \{ x \} ) \cap \{ x_{n} : n \in \mathbb{N} \} \end{align}$ is a subset of an at-most countable set ($x_{n} : n \in \mathbb{N}$) and is hence at most countable. > Let $(x_{n})_{n \in \mathbb{N}}$ be a [[sequence]] of real numbers. $\to.$ Suppose $(x_{n}) \to x$ wrt $\tau_{1}$. Then for every subset $U$ of $\mathbb{R}$ containing $x$ there exists $N \in \mathbb{N}$ s.t. for all $n > \mathbb{N}$ we have $x_{n} \in U$. In particular, choosing $U=\{ x \}$ tells us that if $(x_{n}) \to x$ then there is $N$ large enough that $x_{n}=x$ for all $n > N$... i.e., $(x_{n})$ 'eventually becomes constant'. Such a [[sequence]] [[converge|converges]] wrt any [[topological space|topology]]. In particular, $(x_{n})$ converges wrt $\tau_{2}$. > $\leftarrow.$ Conversely, suppose $(x_{n} \to x)$ wrt $\tau_{2}$. Then for any $U \subset \mathbb{R}$ containing $x$ with $\mathbb{R} - U = \{ y_{n} : n \in \mathbb{N} \}$ for some [[sequence]] $(y_{n})_{n \in \mathbb{N}}$ of reals, there exists $N \in \mathbb{N}$ s.t. $x_{n} \in U$ for all $n > N$. Using the **lemma**, we can in particular, given $U:=\{ x \} \cup (\mathbb{R} - \{ x_{n} : n \in \mathbb{N} \}),$ fix $N \in \mathbb{N}$ s.t. for all $n > N$ we have $x_{n} \in U$. But that enforces $x_{n} \in \{ x \}$, i.e., $x_{n} = x$ for all $n > N$. Hence $(x_{n})$ eventually becomes constant, and in light of the "eventually-constant" characterization of convergence of sequences wrt the [[discrete topology]] given above, we may conclude that $(x_{n})$ converges wrt the [[discrete topology]]. ^fe5948 ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```