topological space (in contrast, only linear subsets of a linear space are linear spaces). Arbitrary topological spaces, investigated by general topology (called also point-set topology) are too diverse for a complete classification up to homeomorphism. Compact topological spaces are an important class of topological spaces ("species" of this "type"). Every continuous function is bounded on such space. The closed interval [0,1] and the extended real line [−∞,∞] are compact; the open interval (0,1) and the line (−∞,∞) are not. Geometric topology investigates manifolds (another "species" of this "type"); these are topological spaces locally homeomorphic to Euclidean spaces (and satisfying a few extra conditions). Low-dimensional manifolds are completely classified up to homeomorphism. Both the linear and topological structures underlie the linear topological space (in other words, topological vector space) structure. A linear topological space is both a real or complex linear space and a topological space, such that the linear operations are continuous. So a linear space that is also topological is not in general a linear topological space. Every finite-dimensional real or complex linear space is a linear topological space in the sense that it carries one and only one topology that makes it a linear topological space. The two structures, "finite-dimensional real or complex linear space" and "finite-dimensional linear topological space", are thus equivalent, that is, mutually underlying. Accordingly, every invertible linear transformation of a finite-dimensional linear topological space is a homeomorphism. The three notions of dimension (one algebraic and two topological) agree for finite-dimensional real linear spaces. In infinite-dimensional spaces, however, different topologies can conform to a given linear structure, and invertible linear transformations are generally not homeomorphisms. Affine and projective spaces It is convenient to introduce affine and projective spaces by means of linear spaces, as follows. A linear subspace of a linear space, being itself a linear space, is not homogeneous; it contains a special point, the origin. Shifting it by a vector external to it, one obtains a affine subspace. It is homogeneous. An affine space need not be included into a linear space, but is isomorphic to an affine subspace of a linear space. All affine spaces are mutually isomorphic. In the words of John Baez, "an affine space is a vector space that's forgotten its origin". In particular, every linear space is also an affine space. Given an affine subspace A in a linear space L, a straight line in A may be