Is cofinite topology first countable?

The set of real numbers with the cofinite topology is not a first countable space. The cofinite topology on a set contains and the complements of finite sets. Suppose is a first countable space. Let be a countable open base at Each is open so is closed hence finite.

Is indiscrete topology first countable?

A topological space (X, T ) is called first countable if each point x ∈ X has a countable neighbourhood basis. This means, for each x ∈ X, there is a countable collection {Ni}i∈N of neighbourhoods of x with the property that, if N is any neighbourhood of x, then there is some i such that Ni ⊆ N.

Is first countability a topological property?

Definition: a first countable space is a topological space in which there exist a countable local base at each of its point. Example : every discrete space is first countable space. Example : (R,U) is first axiom space.

Is topology countable or uncountable?

Since every set is closed if and only if it is finite, there are countably many finite closed sets, and therefore finitely many open sets (the complements of closed sets). So the entire topology is a countable basis for itself, as needed.

What is first-countable topological space?

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the “first axiom of countability”. Specifically, a space is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point in there exists a sequence.

What is the first-countable space in topology?

In other words, a topological space (X,τ) is said to be the first countable space if every point x of X has a countable neighbohood base. A first countable space is also said to be a space satisfying the first axiom of countability. Example: If X is finite, then (X,τ) is first countable space.

What is countable topology?

The cocountable topology or countable complement topology on any set X consists of the empty set and all cocountable subsets of X, that is all sets whose complement in X is countable. It follows that the only closed subsets are X and the countable subsets of X.

Is second-countable a topological property?

Second-countability implies certain other topological properties. Specifically, every second-countable space is separable (has a countable dense subset) and Lindelöf (every open cover has a countable subcover). The reverse implications do not hold.

Is every first-countable space is second countable?

Thus, if one has a countable base for a topology then one has a countable local base at every point, and hence every second-countable space is also a first-countable space. However any uncountable discrete space is first-countable but not second-countable.

Is compact space first-countable?

A space X is first countable if every point has a countable local base; separable if X has a countable dense set; locally compact if every point has a compact neighborhood; and zero-dimensional if every point has a local base of clopen sets.

Is RL second-countable?

If x = y then Bx = By (since x = inf(Bx) and y = inf By). So the mapping x → Bx of Rl onto B is one to one and hence |B| = |Rl| and B is uncountable. That is, Rl is not second-countable.