Continuous

Definition

We say $f$ is continuous (at every $x$) if $f^{-1}(A)\equiv \{x\in X|f(x)\in A\}$ is open for every open $A\subset Y$.

We say $f$ is open if $f(B)$ is open and $\forall$ open $B\subset X$.

If $f$ is a bijection that is both continuous and open, it is a homeomorphism, and $X$ and $Y$ are homeomorphic, written $X\simeq Y$; they are the ‘same’ as topological spaces.

In-depth Definition

A function $f:X\to Y$ between topological spaces is continuous at $x\in X$ if for every neighbourhood $A$ of $f(x)$, we can find a neighbourhood $B$ of $x$ such that $f(B)\subset A$, or $B\subset f^{-1}(A)$.