Is 10 closer to infinity than 1? This may be considered a philosophical question but is the number "10" closer to infinity than the number "1"?
There is more than one notion of "closer than."

Metric: We are not able to extend the metric on $\Bbb R$ to the extended line $[-\infty,\infty]$ in a meaningful way (i.e. where bigger elements of $\Bbb R$ are closer to $+\infty$ than smaller elements). Even if we extend the concept of "metric" to include infinite distances, this doesn't work.

Order-theoretic: A partial order $\le$ on a set is a relation satisfying certain properties. It is a way of ordering the elements of the set, put simply. A linear ordering is one in which every two elements are comparable: for any $x$ and $y$, one of $x\le y$ or $y\le x$ is true at least. The extended real line gets its linear ordering from $\Bbb R$, with $+\infty$ made maximal and $-\infty$ made minimal. If $x,y\le z$ then we can say that $y$ is closer than $x$ to $z$ if $x\le y\le z$ and $x\ne y$. On this view, $10$ is indeed closer to $+\infty$ than $1$ (and we can reverse directions to say $1$ is closer to $-\infty$ than $10$).

One observation to be made here is that we essentially decreed larger numbers closer to $+\infty$ purely by fiat. Rather than making this whole discussion trivial, though, it actually illustrates the power of modern algebra in converting imagination into (mathematical) reality. It makes intuitive sense for bigger numbers to be closer to $+\infty$, and there is a consistent way to encode this mathematically, so we simply make it so and therefore have what we want.

Topological: A metric on a space induces a topology on that space. In particular, a metrizable topology. There are more general topologies though, so we can do and allow more things. However not everything can be compared in terms of "closer than." One possible interpretation is that $x$ is closer than $y$ to $z$ if neighborhoods of $z$ containing $y$ also contain $x$ but not vice-versa. It is generally too optimistic to expect elements to be comparable in this way, but the topology on $\Bbb R$ and the extended line is nice enough to make this an almost useful concept.

The problem with this approach is that neigborhoods can be "disconnected," so we can simply put a neighborhood around $+\infty$ containing both $1$ and $10$ and then delete a patch around $10$, thus telling us that neither $1$ nor $10$ is closer than the other. But if we strengthen our notion to only connected neighborhoods, then we get what we want: every connected neighborhood of $+\infty$ containing $1$ must also contain $10$, but not vice-versa. [This originally slipped my mind, but Mike helpfully pointed it out.]

It is worth noting that the topology on the extended line is induced by the ordering on it (topologies are induced by orderings by considering "intervals" to be a topological base). So this actually isn't fundamentally any newer information than the previous context. Furthermore, different metrics can induce the same topology, and it is possible to make $[-\infty,+\infty]$ a metric space in which $10$ is closer than $1$ to $+\infty$. For example, $d(a,b):=|\tan^{-1}(a)-\tan^{-1}(b)|$ does this, as Rahul points out
My thought is that it is closer to both, dependent on your origin of view. To better define this question we would need a third variable, say PoV for Point of View.

Take an infinite length (aka: a Line) in a straight line. We must therefore assume there is no starting point as this fundamentally destroys our idea of 'infinite'. So where on this infinite length do we place the 1 and the 10? They must be relative to each other given our laws of mathematics i.e. 10 comes after 1 and both are equal distance either side of 5 and so on.

But either side of 10 and 1 is infinite and so infinite has two ends (in the case of a line) and therefore is the same distance to both -infinity (meters or whatever).

Now this raises another question. if we are measuring in a finite length on an infinite line then how do we decide where 1 and 10 must be placed. It could be placed at any length and at any distance apart and the line either side must always be infinite. So then, could the distance between the two points be infinite? Yes.

Personally I think the word 'infinite' is a cause of unnecessary philosophical questions :D

I also think The word infinite should be labelled as a vector or mathematical expression, rather than a number which most people treat it so. 1 and 10 are plotted points on a virtual graph whereas infinity is a mathematically defined expression (whether it be a curve or a straight line is irrelevant).

My 2 cents. Take it with a grain of salt as I am not a mathematician; I'm just a software developer!