szeparálható

Szeparálható is the Hungarian term for “separable.” In topology, a topological space X is szeparálható if it contains a countable dense subset. A subset D is dense in X when every non-empty open set in X intersects D, equivalently when the closure of D is X.

In metric spaces, separability means there exists a countable dense subset. Many standard spaces are separable: for example, the real numbers R with the usual metric are separable because the rationals Q form a dense, countable subset. Consequently, spaces like R^n and function spaces such as l^2 or C([0,1]) are separable as well (polynomials with rational coefficients give a countable dense subset in C([0,1])).

Key properties include: continuous images of separable spaces are separable; closed subspaces of separable spaces are separable; finite or countable products of separable spaces are separable. However, uncountable products of nontrivial spaces generally fail to be separable (for instance, R^I is not separable when I is uncountable).

Related notions extend beyond topology. In functional analysis and operator theory, a C*-algebra is called separable if it has a countable dense subset. In measure theory and probability, separability often appears as a convenient structural condition allowing the use of countable methods.

Examples and non-examples help to illustrate the concept: a discrete space with an uncountable set of points is not separable, while familiar spaces such as R, R^n, l^2, and the space of continuous functions on [0,1] are examples of separable spaces.

In Hungarian mathematical literature, szeparálható is thus typically used to describe spaces or structures that admit a countable dense subset, enabling certain countability-driven techniques and results.