Ttyjfyk

My question is: if you prove that something is true for all points in a topological space or a subset of some topological space, does that imply that this property holds for the whole topological space or the subset of the topological space?

EDIT: more concrete if you have a topological space where all of its points are closed then is this space also closed? If that even makes sense.

If this is true am I then allowed to pick an arbitrary point of the space and then show that since it holds for this one point then the topological space has this property?

The answer, as far as your specific example is concerned, is negative. Every topological space $X$ is a closed subset of itself. However, there are topological spaces in which not all points are closed.

A better example would be: a set which consists of a single point is always compact and connected, but lots of topological spaces are neither compact nor connected.

As your example property "is closed" illustrates, the properties of single points in a space and the space as a whole are not entirely linked together. At the very least, it's not something you can count on in general. I would personally suggest you instead as a general rule assume they are not connected, and make note of the times it does happen.

Taken literally, your question is ill-posed.

This is because a topological space and a point in the topological space are different kinds of things. When we say that a point is closed in a topological space, what we really mean is that its singleton is closed. This is literally very different, but because "a point is closed" taken literally is, in general, nonsensical, this short of shorthand is acceptable.

Having this in mind, you could rephrase your question to a more meaningful (not nonsensical) one:

Let $X$ be a topological space, and let $Asubseteq X$. If $P$ is a topological property of a subset of $X$ and for every $ain A$, the singleton $a$ has the property $P$, does $A$ also have the property $P$?

The answer is trivially no. If you consider "not being a singleton" a topological property, then it fails spectacularly. Otherwise, the property you consider, "being closed" (definitely a topological property) also fails: for example, if you consider $(0,1)subseteq mathbf R$, then (the singleton of) every point in $(0,1)$ is closed in the reals, but $(0,1)$ is not.

You might ask for what properties $P$ this is true. One such property is being open: if for every $ain A$, the singleton $a$ is open in $X$, then $A$ itself is open (as a union of open sets). I strongly suspect that this is just about the only interesting and nontrivial property for which this is true (for suitable notions of "interesting" and "nontrivial").

A related, far more interesting question is about what topological properties are local, or in other words, what properties of a topological space are true for a space if and only if every point has a neighbourhood with the same property.