Continuity is perhaps one of the most fundamental topics in all of analysis, as it provides a way of specifying mappings that are nicely behaved and suitable for studying. Many topics in mathematics (i.e. a homeomorphism) rely on the underlying notion of continuity. Continuous functions have rather nice properties, such as being Riemann and Lebesgue Integrable, mapping compact sets into compact sets, and being uniquely determined by their values on a dense subset.

I begin with the introduction of a metric space, discuss continuity in these spaces, and thus provide motivation for the development of topological spaces.

**Metric Spaces**

Definition 1:Let be a set. AMetric Spaceis simply a pair where is known as themetric functionso that and satisfies the following conditions for :

- is non-negative, with if and only if .
- is symmetric:
- (Triangle Inequality)

Thus, a metric space is simply a space where there is a well-defined notion of “how close” points are, so that the points are topologically distinguishable (in fact, a metric space is Hausdorff).

For example, we may equip with a metric so that is a metric space. Often, we will use one of the following two metrics for :

called the Euclidean Metric (as it is derived from inductively applying the Pythagorean Theorem). As well as the supremum metric:

since is finite-dimensional.

The ambiguity as to *which *metric we should assign to a space therefore arises, as these seem to be entirely different ways of equipping a space with a metric. Consequently, we have the following definition:

Definition 2:Let and be metric spaces. We define an equivalence relation if for all and for some .

Now, it may be (simple exercise) shown that . Thus, there is a notion of equivalence amongst metrics, and so we may interchange them without losing essential information about these sets.

Another basic, though less obvious, example of a metric is the discrete metric. This is the metric space where is defined to be . As we’ll see in a moment, this is an interesting metric space, as all functions with this function as domain will be continuous.

Metric spaces also have notions of *open and closed *sets. Define an *open* *ball *(or *neighborhood*) of radius centered at to be the set . (Equivalently, define a *closed* *ball *to be the set ). Using these, we assume the reader has knowledge of open and closed sets in a metric space. Namely, is open in if for each there exists so that . Equivalently, is closed in if contains its limit points. If the reader is not familiar with this notion, check the notion of convergence in a Metric Space.

**Continuity in Metric Spaces**

We now introduce the notion of continuity in metric spaces:

Definition 3:Let and be metric spaces. We say that iscontinuous atif, for every , there exists so that if then . If is continuous at each , we say that is continuous on .

Intuitively, this means that if two points in are “close enough” we can ensure that two points in the co-domain of are also “close”.

In fact, a metric is itself a continuous function. Fix . We shall show first is continuous as a function . We have if we apply the triangle inequality. Thus, choose and we see that if then , so that is continuous. Now, reverse the roles of and . Combining these facts, we see that, for , where and ,

by the triangle in equality we therefore have

so that the metric function is indeed continuous.

Another important form of a continuous function (in fact, it is uniformly continuous) is that of a *Lipschitz *function. This is a function such that . To see continuity, choose . For those familiar with the notion, it should be noted that bijectivity and continuity do not ensure that a mapping is a homeomorphism (a mapping so that is continuous, bijective, and its inverse continuous). To see this, let and be the metric spaces equipped with the Euclidean and Discrete metics, respectively. Then, the inclusion map

is continuous and bijective, yet

fails to be continuous.

**Characterizations of Continuity in Metric Spaces**

A nice property of metric spaces is the fact that we may characterize the notion of continuity of a mapping in four different, yet equivalent ways.

Theorem 1:Let and be metric spaces. Then, the following are equivalent statements:

- is continuous at .
- requirement of Definition 3.
- Suppose is open in . Then, is open in .
- Suppose is closed in . Then, is closed in .
- (requires notion of convergence) If , then .

Proof:(1) (2) by definition. To see (2) (3), we note that is precisely the definition of the open ball centered at with radius . Thus, we have so that so that the inverse image is open. To see (3) (4), we simply take complements. A set is closed if and only if its complement is open. Thus, is open and so is open. Thus, is closed. The other directions as well as (5) are left to the reader as simple exercises.

**Continuity as a Motivation for Topological Spaces**

Suppose we wish to move away from the notion of a metric space to a more general space, called a *topological space.* I want my new space to be one in which there is a well-defined notion of continuity. How then, should I define this new space? Well, why not make use of (3) in Theorem 1? This gives that the only notion required for continuity is that inverse images of open sets are open. Thus, it makes sense for our new space to have some sense of an open set. We then ask: How do open sets* behave* in ? Let be the collection of all open subsets of . Then, the following can be easily checked:

- , .
- For a finite index set , and open sets , .
- For an arbitrary index set , and open sets ,

Ahh, you say. Why not then define a space that behaves in this manner? A collection of sets that we will call *open* sets that behave in the same way does. Well, this is precisely the motivation behind the definition of a topology.

Let be a set. We define a

topologyon to be a collection of sets whose members we will call open sets, satisfying:

- , .
- For a finite index set , and open sets , .
- For an arbitrary index set , and open sets ,
We call a topological space.

This gives a natural definition of continuity in these spaces:

Definition 4:Let and be topological spaces. Let . We say that is continuous at if, for every open neighborhood of , there exists an open neighborhood of so that whenever . If is continuous at every , we say that is continuous on .

That’s all for now, hopefully this entry gives some insight as to the motivation behind topological spaces, as well as an introduction of the basic concept of continuity. As usual, comments are welcome and all inquiries should be directed to erdosninth@gmail.com

Cheers,

J.T. (S.D.)

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