3.5 Continuity

Definition 3.5.1 (Continuity).label Let $X,Y$ be a topological spaces, $f:X\to Y$ be a function and $x\in X$, then the following are equivalent:

1. (1)

   For each $V\in\cali{N}(f(x)),f\inv(V)\in\cali{N}(x)$.

2. (2)

   For each filter base $\scr{B}\suf \cali{P}(X)$ converging to $x$, $f(\scr{B})$ converges to $f(x)$.

If the above holds, then $f$ is continuous at $x\in X$. Furthermore the following are equivalent:

1. (1)

   For each $U\suf Y$ open, $f\inv(U)$ is open in $X$.

2. (2)

   $f$ is continuous at every $x\in X$.

3. (3)

   For each convergent filter base $\scr{B}\suf\cali{P}(X)$, $f(\scr{B})$ is convergent.

If the above holds, then $f$ is continuous. The collection $C(X;Y)$ is the space of all continuous functions from $X$ to $Y$.

Lemma 3.5.2 (Gluing Lemma for Continuous Functions).label Let $X,Y$ be topological spaces and $\curl{U_i}_{i\in I}$ a countable open covering of $X$. Given functions $\curl{f_i}_{i\in I}$ satisfying:

1. (1)

   For each $i\in I$, $f_{i}\in C(U_{i};Y)$

2. (2)

   For each $i,j\in I$ such that $U_{i}\cap U_{j}\neq\emp$, $f_{i}|_{U_i\cap U_j}=f_{j}|_{U_i\cap U_j}$

then there uniquely exists $f\in C(X;Y)$ such that $f|_{U_i}=f_{i}$ for all $i\in I$.