Are black holes really that special? Science and science fiction alikeScience fiction describes black holes as these amazingly different entities in space that don't behave according to the same laws of physics that the rest of the universe is bound to. I've heard them described as wormholes to other universes, singularities, or tears in space-time.
However, are they really that special? 
My thought is that all matter has event horizons. However, most of it isn't dense enough for that event horizon to be large enough to affect the way the matter interacts with the rest of the universe. So, a black hole is simply just a dense star that is simply dark. 
And, if black holes must have a singularity at their center because of general relativity, can't we conclude from that that all matter creates singularities, since the gravitational field intensifies into infinity the closer you get to the point-mass? What is the difference being really close to a point-mass and being very close to a black hole?
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Science and science fiction alike describe black holes as these amazingly different entities in space that don't behave according to the same laws of physics that the rest of the universe is bound to. I've heard them described as wormholes to other universes, singularities, or tears in space-time.

Um... no. Science fiction may do that, but not science, at least not when it's properly described. Admittedly, popular science writers do have a tendency to treat black holes as mystical objects, but they're really not. The general theory of relativity, our best description of gravity, works just as well for black holes as it does for anything else.
To be clear, let me give out a couple of definitions:


*

*A black hole is a region which, once entered, is impossible to escape. This has a precise mathematical definition in general relativity. The theory allows us to calculate the set of all possible paths that a particle can travel on; they're called geodesics. When you have a black hole, all geodesics which pass inside the black hole remain inside the black hole.

*An event horizon is a surface that can only be crossed in one direction. The boundary of a black hole is one example.

*A singularity is a point (or a set of points) at which a theory predicts that some physical quantity becomes infinite. In the case of a black hole, the quantity is the spacetime curvature. In reality, it doesn't mean that the quantity actually becomes infinite, it just means that you need a better theory.



My thought is that all matter has event horizons. However, most of it isn't dense enough for that event horizon to be large enough to affect the way the matter interacts with the rest of the universe. So, a black hole is simply just a dense star that is simply dark.
And, if black holes must have a singularity at their center because of general relativity, can't we conclude from that that all matter creates singularities, since the gravitational field intensifies into infinity the closer you get to the point-mass? What is the difference being really close to a point-mass and being very close to a black hole?

You're partly right, partly possibly right, and partly wrong. Under the current models which treat elementary particles as points of zero size, then yes, they would have tiny event horizons. People have actually investigated the implications of this, whether elementary particles could actually be black holes, and vice versa. As far as I know, it seems unlikely, but we don't particularly care for now because these black holes would be far tinier than anything we can currently detect. (One of the appeals of string theory is that it spreads out the mass of an elementary particle along an extended object, so there isn't necessarily a high enough density to be a black hole.)
However, a black hole is not just a dense, dark star. Yes, it is dense and dark, but the key difference is that a black hole does not allow anything to escape from inside it, i.e. it has an event horizon. That is not the case with a normal star, at least not on a macroscopic scale.
A: "Science and science fiction alike describe black holes as these amazingly different entities in space that don't behave according to the same laws of physics that the rest of the universe is bound to. I've heard them described as wormholes to other universes, singularities, or tears in space-time." - this is not quite correct! Black holes obey the same physical laws as the rest of the observable universe. They are neither worm holes nor gateway to another universe, nor any tears in space-time.
Classical GR does not allow any tear in spacetime. In GR, geometry of the space-time can change but topology can't change. Only string theory allow such tears but those are not necessarily related with black holes.
Black holes are special in the sense that they contain a singularity at their core and nothing can come of it including light except Hawking radiation which are emitted from the vicinity of a black hole. They are clothed. That means black holes can affect the causal structure of spacetime.
Another thing that is special is black holes are ripe for studying quantum gravity since gravity is extreme in that case.
"My thought is that all matter has event horizons." - sorry, but this is not right.
I think, you should study physics a little more carefully. Cheers.
