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The quotient space / where the natural numbers on the real line are identified as a single point is not first countable. [1] However, this space has the property that for any subset A {\displaystyle A} and every element x {\displaystyle x} in the closure of A , {\displaystyle A,} there is a sequence in A {\displaystyle A} converging to x ...
There exists a non-empty, countable subset S of C such that, if x,y ∈ C such that x < y, then there exists z ∈ S such that x < z < y. (separability axiom) C has no first element and no last element. (Unboundedness axiom) These axioms characterize the order type of the real number line.
General topology grew out of a number of areas, most importantly the following: ... Every continuous function is sequentially continuous. If X is a first-countable ...
Every discrete space is first-countable; it is moreover second-countable if and only if it is countable. Every discrete space is totally disconnected. Every non-empty discrete space is second category. Any two discrete spaces with the same cardinality are homeomorphic. Every discrete space is metrizable (by the discrete metric).
In the other direction, the binary expansions of numbers in the half-open interval [,), viewed as sets of positions where the expansion is one, almost give a one-to-one mapping from subsets of a countable set (the set of positions in the expansions) to real numbers, but it fails to be one-to-one for numbers with terminating binary expansions ...
In mathematics and statistics, a quantitative variable may be continuous or discrete if it is typically obtained by measuring or counting, respectively. [1] If it can take on two particular real values such that it can also take on all real values between them (including values that are arbitrarily or infinitesimally close together), the variable is continuous in that interval. [2]
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. [a] Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time ...
First-countable A space is first-countable if every point has a countable local base. Fréchet See T 1. Frontier See Boundary. Full set A compact subset K of the complex plane is called full if its complement is connected. For example, the closed unit disk is full, while the unit circle is not. Functionally separated