# properties of first countability

###### Proposition 1.

Let $X$ be a first countable topological space^{} and $x\mathrm{\in}X$. Then $x\mathrm{\in}\overline{A}$ iff there is a sequence $\mathrm{(}{x}_{i}\mathrm{)}$ in $A$ that converges^{} to $x$.

###### Proof.

One side is true for all topological spaces: if $({x}_{i})$ is in $A$ converging $x$, then for any open set $U$ of $x$, there is some $i$ such that ${x}_{i}\in U$, whence $U\cap A\ne \mathrm{\varnothing}$. As a result, $x\in \overline{A}$.

Conversely, suppose $x\in \overline{A}$. Let $\{{B}_{i}\mid i=1,2,\mathrm{\dots}\}$ be a neighborhood base around $x$. We may as well assume each ${B}_{i}$ open. Next, let

$${N}_{n}:={B}_{1}\cap {B}_{2}\cap \mathrm{\cdots}\cap {B}_{n},$$ |

then we obtain a set of nested open sets containing $x$:

$${N}_{1}\supseteq {N}_{2}\supseteq \mathrm{\cdots}.$$ |

Since each ${N}_{i}$ is open, its intersection^{} with $A$ is non-empty. So we may choose ${x}_{i}\in {N}_{i}\cap A$. We want to show that $({x}_{i})$ converges to $x$. First notice tat for any fixed $j$, ${x}_{i}\in {N}_{j}$ for all $i\ge j$. Pick any open set $U$ containing $x$. Then ${N}_{j}\subseteq {B}_{j}\subseteq U$. Hence ${x}_{i}\in U$ for all $i\ge j$.
∎

From this, we can prove the following corollaries (assuming all spaces involved are first countable):

###### Corollary 1.

$C$ is closed iff every sequence $\mathrm{(}{x}_{i}\mathrm{)}$ in $C$ that converges to $x$ implies that $x\mathrm{\in}C$.

###### Proof.

First, assume $({x}_{i})$ is in a closed set^{} $C$ converging to $x$. Then $x\in \overline{C}$ by the proposition^{} above. As $C$ is closed, we have $x\in \overline{C}=C$.

Conversely, pick any $x\in \overline{C}$. By the proposition above, there is a sequence $({x}_{i})$ in $C$ converging to $x$. By assumption^{} $x\in C$. So $\overline{C}\subseteq C$, which means that $C$ is closed.
∎

###### Corollary 2.

$U$ is open iff every sequence $\mathrm{(}{x}_{i}\mathrm{)}$ that converges to $x\mathrm{\in}U$ is eventually in $U$.

###### Proof.

First, suppose $U$ is open and $({x}_{i})$ converges to $x\in U$. If none of ${x}_{i}$ is in $U$, then all of ${x}_{i}$ is in its complement $X-U$, which is closed. Then by the proposition, $x$ must be in the closure^{} of $X-U$, which is just $X-U$, contradicting the assumption that $x\in U$. Hence ${x}_{i}\in U$ for some $i$.

Conversely, assume the right hand side statement. Suppose $x\notin {U}^{\circ}=X-\overline{X-U}$. Then $x\in \overline{X-U}$. By the proposition, there is a sequence $({x}_{i})$ in $X-U$ converging to $x$. If $x\in U$, then by assumption, $({x}_{i})$ is eventually in $U$, which means ${x}_{i}\in U$ for some $i$, contradicting the earlier statement that $({x}_{i})$ is in $X-U$. Therefore, $x\notin U$, which implies that $U\subseteq {U}^{\circ}$, or $U$ is open. ∎

###### Corollary 3.

A function^{} $f\mathrm{:}X\mathrm{\to}Y$ is continuous^{} iff it preserves converging sequences.

###### Proof.

Suppose first that $f$ is continuous, and $({x}_{i})$ in $X$ converging to $x$. We want to show that $(f({x}_{i}))$ converges to $f(x)$. Let $V$ be an open set containing $f(x)$. So ${f}^{-1}(V)$ is open containing $x$, which implies that there is some $j$ such that ${x}_{i}\in {f}^{-1}(V)$ for all $i\ge j$, or $f({x}_{i})\in V$ for all $i\ge j$, which means that $(f({x}_{i}))\to f(x)$.

Conversely, suppose $f$ preserves converging sequences and $C$ a closed set in $Y$. We want to show that $D:={f}^{-1}(C)$ is closed. Suppose $({x}_{i})$ is a sequence in $D$ converging to $x$. Then $(f({x}_{i}))$ converges to $f(x)$. Since $(f({x}_{i}))$ is in $C$ and $C$ is closed, $f(x)\in C$ by the first corollary above. So $x\in {f}^{-1}(C)=D$ too. Hence $D$ is closed, again by the same corollary. ∎

Title | properties of first countability |
---|---|

Canonical name | PropertiesOfFirstCountability |

Date of creation | 2013-03-22 19:09:26 |

Last modified on | 2013-03-22 19:09:26 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 11 |

Author | CWoo (3771) |

Entry type | Result |

Classification | msc 54D99 |