generated from smyalygames/quartz
8 lines
788 B
Markdown
8 lines
788 B
Markdown
# 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 Sets|open]] if $f(B)$ is [[Open Sets|open]] and $\forall$ [[Open Sets|open]] $B \subset X$.
|
|
|
|
If $f$ is a [[Bijective|bijection]] that is both **continuous** and [[Open Sets|open]], it is a [[Homeomorphic|homeomorphism]], and $X$ and $Y$ are [[Homeomorphic|homeomorphic]], written $X \simeq Y$; they are the 'same' as [[Topological Space|topological spaces]].
|
|
## In-depth Definition
|
|
A function $f : X \to Y$ between [[Topological Space|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 $B \subset f^{-1}(A)$. |