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.
Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s