Open Ball in a Metric Space vs. Open Set in a Topological Space

Question

I'm having trouble understanding the notion of an open set when applied to a space without a metric defined on it. I have read that all metric spaces are naturally a topological space, but the converse is not true. Topological spaces definitions I have read use the idea of open sets, and I can't understand this abstract idea of openness or closedness of a set without having a notion of distance.

I can understand the notion of open and closed sets in a metric space from the definitions I have read using the idea of distance and open balls. So for example given a metric space $M$ with the metric $d$, we can say a set $U \subset M$ is open if $\forall x \in U, \exists B(x,r) \subset U$, which says that we can choose any point within the set $U$ and there will always be some sufficiently small distance $r$ that we can move in any direction to another point $y$ that is also contained within $U$. This has an intuitive conceptual meaning in my head regardless of space $M$ or metric $d$.

Now take the example of a space $F$, a fruit bowl with $3$ apples, $3$ oranges and $2$ bananas. There is no metric defined on the space $F$ to determine a distance between its elements, the fruit. Can we define an open set on this space? In order to be a topological space we need to be able to define open sets right?

Answer 1

Having some notion of open sets lets you define notions of points getting arbitrary close to each other, or continuity. For example in space of real functions you can choose topology (in other words: choose which sets you declare open) such that sequence of functions converges to some given function pointwise. There is no natural notion of distance related to this mode of convergence, yet it is clear that in SOME sense the converging functions are getting closer and closer. There are also other notions on closeness on such a space, for example you can say that sequence converges if convergence is uniform - and then you actually get familiar metric space. To understand topology you need a) to look for examples and b) notice that many notions in metric space theory can be defined only in terms of open sets and you will be able to extend them outside metric spaces if you define open sets abstractly. Then after some practice and experience you will notice that such generalisations are very natural in many contexts and topology is very powerful tool.

Answer 2

Take a set $X$. Let $\lambda \in X$ , we say that a ball of radius epsilon ($\epsilon$) around the point $\lambda$ will be an open set iff $\ni$ a point $k\notin X$ such that a neighborhood of radius delta ($\delta$) around the point $k$ is not in the ball of radius $\epsilon$ around $\lambda$.

In mathematical terms , $B_\epsilon(\lambda)$ is open when,$$N_\delta(k) \cap B_\epsilon(\lambda)=\emptyset.$$

If you want to think in terms of a topological space: We say that let $Y\subset X$ , then given a set $A$ in $Y$ we say that it is open if $A=Y\cap U$ where $U$ is open in $X$

Answer 3

Maybe an analogy will help. Supposing you start with an understanding of Real Analysis, you can generalize this understanding to Metric Spaces:

In Real Analysis, you have this fixed notion of distance between two points in $\mathbb{R}$ defined as $|x-y|$. Then using this definition you build your collections of open sets that you use to study $\mathbb{R}$. These sets look like open intervals $(x-r, x+r)$ centered at $x$ with radius $r$, and using these open sets you can study the space (continuity, connectedness, derivatives, etc.)

Then you jump to into a general a Metric Space and are told "Use whatever definition of distance that you like (as long as it follows a few rules to keep things sensible) and build your open sets with that!" Then you just proceed to do the same thing you did in Real Analysis only you are using open balls that look like $B(x,r)$, centered at $x$ with radius $r$.

Now we can see a related jump from understanding Metric Spaces to understanding General Topology:

In Metric Spaces, you have this fixed notion of distance between two points in your space $X$ defined as $d(x,y)$. Then using this definition you build your collections of open sets that you use to study $X$. These sets look like open balls $B(x,r)$ centered at $x$ with radius $r$, and using these open sets you can study the space (continuity, connectedness, derivatives, etc.)

Then you jump to into General Topology and are told "You don't need to bother building the collection of open sets at all. Instead, I'm just going to hand you this collection of open sets! (and these open sets will follow a few rules to keep things sensible)" Then you just proceed to do the same thing you did with Metric Spaces only you are using these open sets that you were handed to you, and there might not be a easy way to describe what they look like.