If X be a second countable space and N a base for the topology of X . Show that there is a countable base B for the topology of X such that B ⊂ N.
105358
In a second countable space any base has a countable subset that is still a base
16413
Proving existence of surjective $f:\mathbb{N} \rightarrow A$ implies $A$ is at most countable.
119649
Every metrizable space with a countable dense subset has a countable basis
42913
Equivalence of the definition of the Subbasis of a Topology
108864
Showing that two topologies give rise to the same topology
62911
Let $(X,\tau)$ be a connected topological space, $Y\subset X$ connected and $U\subset X\setminus Y$ clopen. Prove that $ Y \cup U$ is connected
116768
Problem 5, sec. 13 in Munkres' Topology, 2nd ed.: How to prove the assertion if $\mathscr{A}$ is a subbasis?
80414
Countable Product of Sequentially Compact spaces is Sequentially Compact
19297
Is the cofinite topology on an uncountable set first countable?
23907
$Seq (\mathbb{N})$ of all finite sequences of elements of $\mathbb{N}$ is countable.
149980
Are there any countable Hausdorff connected spaces?
77563
Homeomorphism between spaces equipped with cofinite topologies
80096
Prove that the union of countably many countable sets is countable.
52429
Explicit countable elementary extension of $\mathbb{N}$
78513
Show finite complement topology is, in fact, a topology
66239
Verifying that a certain collection of intervals of $\mathbb R$ forms a topology
7678
Show "countable complement topology" is a topology