## Motivation and Definition

While studying analysis, one notices that many important concepts can be defined in terms of “open sets”. One gets the inkling that this concept is critical in forming our notions of continuity, limits etc. In this article, we shall abstractise this further by considering only open sets. At first glance, it does seem rather unlikely that such a general definition could be useful in any manner. However, it will soon become clear that abstractisation has the following advantages:

- It brings out the essence of proofs of theorems, by highlighting specific properties of an object which are critical to the proofs.

- The concepts can be applied to vastly different areas of mathematics, including for example, number theory, Galois theory, commuative algebra (in more ways than one!) etc, and have proven to be extraordinarily fruitful in furthering their study.

Enough talk, let’s begin. Let *X* be any set. As mentioned above, we wish to define a topology on it by stating what constitutes “open sets”.

Definition. Let be the power set of X (i.e. set of all subsets of X). Atopologyon X is a subset satisfying:

- ;
- if , then ;
- if is a collection of elements of T, then .
If U is a subset of X belonging to T, we say U is an

open subsetof X. If C is a subset of X whose complement X-C belongs to T, we say C is aclosed subsetof X.

Some observations:

- The definition does not preclude the possibility that a subset can be both open and closed, so the terminology is a little unfortunate. For a quick motivation of why closed subsets are of interest, read here.
- By repeatedly applying intersection, one notes that if are open, then so is . In other words, a topology is closed under an intersection of finitely many terms, but not infinitely many terms in general (see the example of
**R**later, where the intersection of all is {0}, which is not open).

Note that we could also define a topology by stating what constitutes “closed subsets” of X. Thus:

- are closed subsets of
*X*; - if are closed subsets of
*X*, then so is closed; - if is a collection of closed subsets of
*X*, then is closed.

Definition. Suppose are both topologies on X. If , then we say is afiner topologythan and conversely is acoarser topologythan .

[ Note: this diagram is meant as an illustration of finer/coarser; it does not represent actual topologies. ]

## Examples of Topologies

- On any set
*X*, we have the**discrete topology***T*=**P**(*X*) where every subset is open; this is the finest topology possible for*X*. Also, there’s the**trivial topology**, which is the coarsest topology possible for*X*. - Let . The standard topology on
*X*was already defined earlier. Recall that is open if and only if for each*x*in*U*, there is an ε>0, such that any*y*in*X*satisfying |*y*–*x*|<ε must be in*U*also. - Let
*X*be a set. The**cofinite topology**is defined via having open if and only if*X*–*U*is finite. In other words, the class of closed subsets is precisely the class of finite subsets, from which it’s clear that the cofinite topology satisfies the set of axioms of a topology. - Let
**N**= {1, 2, … } be the set of positive integers. The**right order topology**on**N**is where the only non-empty open subsets are for*m*=1, 2, 3, … . Note that and ; the fact that the order topology is closed under arbitrary union follows from that every subset of**N**has a minimum. - If , where ∞ is just a dummy symbol here, then we can define a topology on
**N***by decreeing that the only non-empty open subsets are*Thus, any open subset containing ∞ contains all sufficiently large natural numbers*.

[ Side note: more generally, one can define the order/right-order and left-order topology on any totally ordered set. Note that the right-order topology on **N*** differs from the one described in example 5, since {∞} is an open subset in the right-order topology. ]

## Metric Spaces

One common source of topologies is via a metric function, which provides a geometry by indicating the “distance” between any two points in a set.

Definition. Ametricon a set Xis a function such that:

- d
(x,y) ≥ 0 for anyx,yinX, with equality if and only ifx=y;- d
(x,y) =d(y,x) for anyx,yinX;- d
(x,z) ≤d(x,y) +d(y,z) for anyx,y,zinX(triangular inequality).The pair (X, d) is then called a

metric space.

The triangular inequality ensures that the function *d* behaves like a “distance function”.

**Examples**

- The
**discrete metric**on any*X*is defined by d(*x*,*y*) = 0 if*x*=*y*and 1 otherwise. - Let . The
**Euclidean metric**is the standard distance function . - Let again. This time we take the metrics and . [ See diagram below. ]
- Let
*p*be a prime number and*X*=**Z**, the set of integers. Given any integer*m*let where is the highest power of*p*dividing*m*; in the case*m*=0, since all powers of*p*divide*m*we have . Then is a metric, called the**p****-adic metric**on**Z**. This was briefly alluded to in a prior post. [ Note: the triangular inequality follows from the fact that if divides both*x*–*y*and*y*–*z*, then it divides*x*–*z*. One way of looking at this metric is that**two integers are near if their difference is divisible by a high power of**]*p*.- E.g. suppose
*p*=5. Then*d*(5, 3)=|5-3|_{5}= 0,*d*(57, -18)=|75|_{5}= 1/25, and*d*(152, 27) = 1/125.

- E.g. suppose

The following theorem shows that a metric space gives rise to a topology.

Theorem. Let (X, d) be a metric space. For x in X and ε>0, theopen ballwith radius ε>0 and centre x is defined byWe say that a subset is

openif for any x in U, there is an open ball for some ε>0. Then the collection of open sets form a topology.

[ Note: the above diagrams for *d*, *d*_{1} and *d*_{∞} illustrate the open balls for the respective metrics. ]

**Proof**.

Clearly, and *X* are both open. Now, suppose are open (in the sense of metric space). If , then *x* lies in both *U*_{1} and *U*_{2}. Thus, we can find such that:

and .

So .

Finally for arbitrary union of open subsets, let be open and . If *x* lies in *U*, then it must lie in some *U _{i}*. Thus, for some ε>0, . ♦

For example,

- the above metric space induces the topology on
**R**^{n}defined earlier; - the discrete metric on any set
*X*induces the discrete topology; - the open ball
*N*(*x*, ε) itself is an open subset of*X*(the proof is an exact replica of the one in an earlier post, by replacing |**x**–**y**| with*d*(*x*,*y*)).

We say that a topology (*X*, *T*) is **metrisable** if there’s a metric *d* on *X* which induces it. But even if such a *d* exists, it is usually far from unique. For example, the above metrics *d*, d_{1} and *d*_{∞} on **R**^{n} all induce the same topology as we will show a short while later.

Definition. Two metrics d and d’ on X are said to betopologically equivalentif they induce the same topology.

Theorem. This happens if and only if:

- for any x in X and ε>0, there exists δ>0 such that ;

- for any x in X and ε>0, there exists δ>0 such that ;
[ Since there’re two metrics here, we use the subscript N

_{d}to denote the open ball induced by d. ]

**Proof**

Suppose *d* and *d’* induce topologies *T* and *T’* respectively. It suffices to show that the first condition is equivalent to that *T’* is finer than *T*.

For the forward direction, suppose , i.e. it is open in the metric space (*X*, *d*). Then for any *x* in *U*, for some ε>0. By the first condition, for some δ>0. Hence it follows that *U* is open in the metric space (*X*, *d’*) too.

For the converse, suppose let *x* be in *U* and ε>0. Then since is open in *T*, it is also open in *T’*. Hence for some δ>0. ♦

For example, the following diagram shows that *d*_{1} and *d* are topologically equivalent on **R**^{n}.

Conclusion: every metric space induces a topology and many different metrics on the same space can induce the same topology. We’ll see later that some topologies are inherently non-metrisable.