Non-Metrizable Topological Spaces

Question

What are some motivations/examples of useful non-metrizable topological spaces? I am trying to get a feel for what parts of math have topologies appear naturally, but not induced by a metric space. Also, it would be cool and informative if you could list some basic topological properties that each of these spaces have. Thanks.

Answer 1

$\pi$-Base lists ninety-five non-metrizable spaces. Rather than include them all in this post, I will simply link you to the search result.

Answer 2

1) Co- countable topology 2) Co- finite topology 3) Sierpiński spaces are example of non metrizable topological spaces.

Normally topology just comes from the basis[generator of a topology], but metric spaces come from the distance function $d$. Then we observe that the open balls gives us the basis & the topology generated from that is the metrizable topology.

Again see all the topological properties concerning open sets such as arbit union of open sets are open, convergence of sequence, continuity[with taking distance in the role] etc hold for both spaces metrizable & non metrizable topological spaces , where the difference lies is concerning distance such as Hausdorff property, $T_1,T_3 $ properties etc.

Answer 3

Any topological space which is not Hausdorff is non-metrizable.

Answer 4

Maybe you would like the Zariski topology and things like that.

Answer 5

There are lots of nonmetrizable spaces that are relevant. Not just pathological examples, like the trivial topology.

Wikipedia has a good example: all functions $f:R\rightarrow R$ with the topology of pointwise convergence. Read about it here: What is the Topology of point-wise convergence?.

Answer 6

When I started learning Topology, grasping this motivation was really important to me as well. Why go one level of generalization from metric spaces to topological spaces? There are two reasons I focus on: the mathematical method, and quotient spaces.

Generally in mathematics, we notice patterns in the objects we study and then see how to generalize. The perfect example of this, and one that is simple to grasp, is that of groups. We can study integers, and modular arithmetic and find that the results we hold important are not reliant on the definition of the operator $(\cdot)$, but it's algebraic properties. Similarly, we find that a lot of results of open sets ($e.g.$, results regarding compactness, connectedness, $etc.$) only rely on the behavior of the these sets under specific set operations (unions and intersections). So any set of subsets that satisfy that topological axioms will give the same results for open sets in a metric space. But why stop at these axioms? And are we putting the cart before the horse? Do we have any use for topological space axioms that we cannot do with a metric space? Yes. But first...

The definition of an open set in a metric space X is a set $U$ such that for each $x\in U$, there is an $\epsilon>0$ such that the ball $B_\epsilon(x)\subset U$ (where $B_\epsilon (x) = \{y\in X : d(x,y)<\epsilon\}$, or all elements of $X$ that are within a distance of $\epsilon$ from $x$). This just generalizes to the concept of a basis. There is a collection of subsets (balls) that form the basic building blocks of what it means to be "near" the point $x$. We then can define open set from this basis and learn that the open sets satisfy the topological axioms.

But why not just define a topological space from a basis? It's exactly what we do in metric spaces, after all. Well, the answer is in one of the most important uses of topology, quotient spaces.

The naive concept of quotient spaces comes from partitioning a topological space $X$ into disjoint sets called equivalent classes that cover the whole space. We then consider these classes as its own space. This is effectively gluing a collection of elements of $X$ together to form a single element.

For example, take the unit square $I = [0,1]\times [0,1]$ and a topology such that open sets in $I$ behave as you think they would, every element of an open set $U$ is an interior point of $U$ if interpreted geometrically. Now partition $I$ such that if the point $(x_1, x_2)$ is inside the square, it is partitioned into it's own class $\{(x_1, x_2)\}$. But now, "glue" points on the left and right edges of the square together by putting them in the same class. An example of these classes are $\{(0,x_2), (1,x_2)\}$. These two points can be thought of as now being the same point. To represent this using a diagram, imagine taking a paper square and glueing two opposite edges together to make a tube. This is a diagram to help us, but we are not actually moving any elements geometrically (key component is "metric" there), we are just classifying elements together.

How would we go about defining an open set on these equivalent classes? Well, the natural way to do so is the following. Let $U$ be open in $I$. But require that if $x\in U$ then every other $y$ that is equivalent to $x$ (i.e., in the same partitioned class) is also in $U$. Intuitively, this is saying that $U$ does not "cut in half" any equivalent class, it either includes all of it or none of it. An open set that "cuts in half" elements would be a $U$ that contains all of the right edge but non of the left edge.

We note that these new open sets follow the topological space axioms. Furthermore, we have generated them without using a metric space. And our old concept of a metric on $I$ no longer works. It doesn't mean anything to be $\epsilon$ distance away from a glued edge point; a point on the interior of the square will generally not have the same distance to both of the edge points that have been glued together. AND our concept of a basis doesn't even work anymore. Most "small" basis element $B_\epsilon(x)$ around point $(0,x_2)$ on the edge of $I$ will not include its companion point $(1,x_2)$. Any basis element that does include both edge companion points will be very large. With only those large neighborhoods, we cannot generate the topological space that we defined above. So we are really just left with a new definition of an open set and all we know about it is that it follows the three axioms involving unions and intersections. We are required to use the axioms, for we cannot use anything else (generally).