What is the effect of assigning a sign to $0$? What is the effect of assigning a sign to $0$?  In Calculus, $0^+$ means the right side of $0$ whereas $0^-$ means the left side of $0$ in a limit.
But other than that, what are the effects when we try to give $0$ a sign?
What are the consequences?
 A: Usually for a sign one will want to have that the sign of $x$ is different from the sign of the opposite element to/the additive inverse of $x$. 
Now, what does this mean for $0$. As $0$ is its own additive inverse one will either need to have a special sign for it or assign both signs to it. 
Ultimately both lead to the same. 
Indeed, in some languages, e.g., French, it is common that the positive reals include $0$, and so do the negative reals. 
While in English it is common that neither includes them. 
Yet, note that in the Wikipedia page on ordered fields the element $0$ is in fact an element of the positive cone (though not a positive element).
A: Generally we don't refer to 0 as having any sign. Instead, it gets its own class. You could represent that as a symbol of you really wanted to, but the whole reason we use a negative sign is for simplicity... we could come up with a different numbering system for negative numbers so that no sign would be needed, but it's much easier to just mirror our positive reals to the negative reals with a sign in front. With 0, there is only one case, and it mirrors only itself, making a sign redundant. As a note, whether to include 0 as a natural number had been a long debate, and multiple notations exist to include or exclude it. You even get things like "whole numbers", "natural numbers", "non-negative integers" (the last doesn't include 0, since 0 isn't positive) and many more notations, both written out and symbolic
Edit: I should note that some computer software uses $-0$. I can't remember off the top of my head, but some scripting for web pages differentiate between $0$ and $-0$, and the problem is very important when you are discussing Two's complement vs One's complement (I'll explain this onto you if you want, but there are great tutorials online!)
Tl;dr $-0 =0$ for all intensive purposes
A: I don't understand the details, but this seems to have something to do with filters.

Definition 0. Given a topological space $X$ and an element $x \in X$, define that $x^\star$ is the collection of all neighbourhoods of $x$. This happens to be a filter, called the neighborhood filter of $X$.
Definition 1. Given a totally ordered set $X$ and an element $x \in X$, it should be possible to define $x^+$ and $x^-$ to be $x^*$ for two different ways of making $X$ into a topological space.

It should then be possible to define notions like "right-continuous" in terms of limits that make reference to $x^+$ and $x^-$. Perhaps we can even make the set $$\{x \mid x:\mathbb{R}\} \sqcup \{x^+ \mid x:\mathbb{R}\} \sqcup \{x^- \mid x:\mathbb{R}\} \sqcup \{x^\star \mid x:\mathbb{R}\}$$ into an algebraic structure in its own right. So for example, I imagine that we should have:
$$2 \cdot (3^+) = 6^+, \qquad (-2) \cdot 3^+ = (-6)^-$$
etc.
I've made this answer community wiki; can someone who understands the details spell them out here? This would help me, too.
