# An example of a quotient space, I

It is well known that in a compact Hausdorff space, if every point in the space is a $G_\delta$ point, then the space is first countable (see The cardinality of compact first countable spaces, II). Outside of compact spaces, this is not true. We present an example of a space where every point is a $G_\delta$ point but has no countable local base at any one point. This is an opportunity to introduce the notion of quotient space. Our example can be viewed as the image of a first countable space under a quotient map. We first define the example and then present a brief discussion of quotient spaces.

Let $\mathbb{N}$ be the set of all positive integers. Let $W=\left\{\frac{1}{n}: n \in \mathbb{N}\right\}$. Let $S_1=\mathbb{N} \times W$ and $S_2=\mathbb{N} \times \left\{0\right\}$. Consider $X=S_1 \cup S_2$ as a subspace of the Eucliean plane. In the Euclidean topology, each point is $S_1$ is isolated and an open neighborhood at each point $(i,0)$ in $S_2$ is of the form:

$\left\{(i,0)\right\} \cup \left\{(i,\frac{1}{k}):k \ge n\right\}$ for some $n \in \mathbb{N}$

We use $X$ to define a space $Y$ by considering $S_2$ as one point (call this point $p$). Thus $Y=S_1 \cup \left\{p\right\}$. Points in $S_1$ remain isolated. An open neighborhood of $p$ is of the form $\left\{p\right\} \cup U^-$ where $U^-=U-S_2$ and $U$ is a Euclidean open subset of $X$ such that $S_2 \subset U$.

To facilitate the discussion, we describe the open sets at the point $p$ in more details. Let $\mathbb{N}^{\mathbb{N}}$ be the set of all functions $f:\mathbb{N} \mapsto \mathbb{N}$. For each $i \in \mathbb{N}$ and $j \in \mathbb{N}$, let $V_{i,j}=\left\{(i,\frac{1}{k}): k \ge j\right\}$. An open neighborhood of $p$ is of the following form:

$B_f=\left\{p\right\} \cup \bigcup \limits_{i=1}^{\infty} V_{i,f(i)}$

To describe in words, each open neighborhood at $p$ is obtained by removing finitely many points in each vertical segment $V_{i,1}$.

The resulting space $Y$ is not first countable. We show that any countable set of open neighborhoods at $p$ cannot be a local base at $p$. To this end, we consider $B_{f_1},B_{f_2},B_{f_3},\cdots$ where each $f_i \in \mathbb{N}^{\mathbb{N}}$. We wish to find an open set $O$ such that $p \in O$ and each $B_{f_i}$ is not a subset of $O$.

For each $i \in \mathbb{N}$, pick some $g(i) \in \mathbb{N}$ such that $g(i)>f_i(i)$. Consider $O=B_g$. Note that for each $i \in \mathbb{N}$, $(i,\frac{1}{f_i(i)}) \in B_{f_i}$ and $(i,\frac{1}{f_i(i)}) \notin O=B_g$. This shows that no countable number of open neighborhoods can form a base at the point $p$.

However, every point in the space $Y$ is a $G_\delta$ point. For the point $p$, we have $\left\{p\right\}=\bigcap \limits_{i=1}^\infty B_{h_i}$ where $h_i \in \mathbb{N}^{\mathbb{N}}$ is a constant function mapping to $i$.

Quotient Spaces
We now present one definition of quotient spaces. Let $(X,\tau)$ be a topological space. Let $q:X \mapsto Y$ be a surjection, i.e. $q(X)=Y$. Consider the following collection of subsets of $Y$:

$\tau_q=\left\{O \subset Y: q^{-1}(O) \text{ is open in }X\right\}$

The set $\tau_q$ is a topology for the set $Y$. With this topology $\tau_q$ on $Y$, the function $q$ is continuous. In fact, $\tau_q$ is the finest (largest or strongest) topology on $Y$ that makes the function $q$ continuous, i.e. if $\tau_1$ is another topology on $Y$ making $q$ a continuous function, then $\tau_1 \subset \tau_q$.

The topology $\tau_q$ is called the quotient topology induced on $Y$ by the mapping $q$. When $Y$ is given such a quotient topology, it is called a quotient space of $X$ (most of the times just called quotient space). The induced map $q$ is called a quotient map. If $Y$ has a quotient topology defined by a quotient map, then $Y$ is said to be the quotient image of $X$. We plan to discuss quotient space and quotient topology in greater details in this blog. For further information, see [2].

In the example discussed in this post, the quotient map is to map each point in $S_1$ to itself and to map each point in $S_2$ to a single point $p$. In essence, we collapse the whole x-axis $S_2$ into one single point. The open sets for the point $p$ are simply the Euclidean open sets containing $S_2$. This example shows that the quotient image of a first countable space needs not be first countable. Though the quotient image of a first countable space may not be first countable, it has the property that sequences suffice to define the topology (these are called sequential spaces, see [1]).

Reference

1. Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
2. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.