A black hole is a region of spacetime where gravity prevents anything, including light, from escaping.The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizonthat marks the point of no return. It is called "black" because it absorbs all the light that hits the horizon, reflecting nothing, just like a perfect black body in thermodynamics.[ Quantum mechanics predicts that black holes emitradiation like a black body with a finite temperature. This temperature is inversely proportional to the mass of the black hole, making it difficult to observe this radiation for black holes of stellar mass or greater.
A topological space X is a normal space if, given any disjoint closed sets E and F, there are open neighbourhoods U of E and V of Fthat are also disjoint. More intuitively, this condition says that E and F can be separated by neighbourhoods.
The closed sets E and F, here represented by closed disks on opposite sides of the picture, are separated by their respective neighbourhoods U and V, here represented by larger, but still disjoint, open disks.A T4 space is a T1 space X that is normal; this is equivalent to X being Hausdorff and normal.
A completely normal space or a hereditarily normal space is a topological space X such that every subspace of X with subspace topology is a normal space. It turns out that X is completely normal if and only if every two separated sets can be separated by neighbourhoods.
A completely T4 space, or T5 space is a completely normal Hausdorff topological space X; equivalently, every subspace of X must be a T4 space.
A perfectly normal space is a topological space X in which every two disjoint non-empty closed sets E and F can be precisely separated by a continuous function f from X to the real line R: the preimages of {0} and {1} under f are, respectively, E and F. (In this definition, the real line can be replaced with the unit interval [0,1].)
It turns out that X is perfectly normal if and only if X is normal and every closed set is a Gδset. Equivalently, X is perfectly normal if and only if every closed set is a zero set. Every perfectly normal space is automatically completely normal.
A Hausdorff perfectly normal space X is a T6 space, or perfectly T4 space.
Note that the terms "normal space" and "T4" and derived concepts occasionally have a different meaning. (Nonetheless, "T5" always means the same as "completely T4", whatever that may be.) The definitions given here are the ones usually used today. For more on this issue, see History of the separation axioms.
Terms like "normal regular space" and "normal Hausdorff space" also turn up in the literature — they simply mean that the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space is the same thing as a T4 space. Given the historical confusion of the meaning of the terms, verbal descriptions when applicable are helpful, that is, "normal Hausdorff" instead of "T4", or "completely normal Hausdorff" instead of "T5".
Fully normal spaces and fully T4 spaces are discussed elsewhere; they are related to paracompactness.
A locally normal space is a topological space where every point has an open neighbourhood that is normal. Every normal space is locally normal, but the converse is not true. A classical example of a completely regular locally normal space that is not normal is the Nemytskii plane.
