
$U$ is an open neighborhood of $x$:
\\
$U$ is a neighborhood of $x$ and $ U$ is an open set of $X$

Open neighborhood


$U$が$x$ is an open neghborhood $\Leftrightarrow$ $x\in U \land U\in \mathcal{O}_X$

Corollary


$f$ is continuous at $x$:
\\
 $\forall$ open neighborhood of $ f(x)$, $\exists U$: open neighborhood of $ x, (f(U)\subset V)$

Continuous

$f$ is continuous:$\forall x\in X (f$ is continuous at $x$$)$

Continuous map

$\begin{aligned}
 & 1. f\text{ is continuous}
\\ 
 & 2. \forall V (f^{-1}(V)\text{ is open set of }X)
\\
& 3. \forall V' (f^{-1}(V)\text{ is close set of }X)
\end{aligned}$



$f,g$ are continuous $\Rightarrow g\circ f$ is continuous

$f$ is Homeomorphism between$X,Y$:
\\ 
$\begin{aligned}
 & 1. f\text{ is bijective}
\\ 
 & 2. f,f^{-1}\text{ are continuous}
\end{aligned}$

Homeomorphism


$(X,\mathcal{O}_X)$ and $(Y,\mathcal{O}_Y)$ are homeomorphic:
\\
homeomorphism between $X,Y$ exists

Continuous map

$Open neighborhood$

$Corollary$

$Continuous$

$Continuous map$

$Corollary$

$Corollary$

$Homeomorphism$

$Homeomorphic$