Topological space and continuous maps
continuous
$f$ is continuous at $x_0$ if for any open subset $V$ containing $f(x_0)$, then there is a open subset $U$ containing $x_0$ such that $f(U)\subset V$
Definition:
$X$ is any set, the collection of all subsets of $X$ is a topology, called the discrete topology.
Definition:
Let $X$ be a set, and $\tau_f$ be the collection of all subsets $U$ of $X$ such that $X-U$ either is finite or is all of $X$, $\tau_f$ is called the finite complement topology.
Proposition
If $A,B$ are the complement of each other, then $\overline{A},\r{B}$ are the complement of each other
Definition
If a topological space $X$ has a countable dense subset. then $X$ is called separable
Proposition
$C\subset A\subset X$ in a topological space $X$, then $C$ is closed in $A\iff C$ is the intersection of $A$ and a closed subset of $X$
Exercise
$A$ is dense in a topological space $X\iff$ the nonempty open subset intersects with $A$
Definition:
If $X$ is a set, a basis for a topology on $X$ is a collection $\mathcal{B}$ of subsets of $X$ such that (1) For each $x\in X$, there is at least one basis element $B$ containing $x$; (2) If $x$ belongs to the intersection of two basis elements $B_1$ and $B_2$, then there is a basis element $B_3$ containing $x$ such that $B_3\subset B_1\cap B_2$.