📂Topology

Continuity in Topology

Definition

Korean

For topological spaces $(X, \mathscr{T}_{X} )$ and $(Y, \mathscr{T}_{Y} )$, let $f: X \to Y$. If for every $V \in \mathscr{T}_{Y}$ containing $f(a)$ there exists $U \in \mathscr{T}_{X}$ containing $a$ such that $f(U) \subset V$, then $f$ is said to be continuous at $a$. If $f$ is continuous at every point of $X$, it is called a continuous function and can be denoted by $f \in C(X,Y)$.

English

$f$ is continuous at $a$ $\iff$ For all neighborhood $V \in \mathscr{T}_{Y}$ of $f(a)$, there exists a neighborhood $ U \in \mathscr{T}_{X}$ of $a$ such that $a \in U \implies f(a) \in f(U) \subset V$

Explanation

When you first see this definition it may be hard to understand what it means, but if you think about it carefully you can see that it is exactly the same sense as when defining continuity in analysis, where for every given $\epsilon > 0$ we assert the existence of a $\delta$ such that $\left| x - a \right| \lt \delta \implies \left| f(x) - f(a) \right| \lt \epsilon$. Noting that $\left\{ x : \left| x - a \right| \lt \delta \right\}$ and $\left\{ f(x) : \left| f(x) - f(a) \right| \lt \epsilon \right\}$ are open sets, you will understand that saying "a $\delta$ can be found for every given $\epsilon$" is the same as saying "an open set in $X$ satisfying the condition can be found for every open set given in $Y$."

For reference, $C(X,Y)$ is the set of continuous functions whose domain is $X$ and codomain is $Y$. If you are studying topology at this level, you have probably seen the epsilon-delta argument worn out over and over, so formulas and symbols will be more comfortable than prose.

Continuity has been generalized beyond Euclidean space to metric spaces, and now beyond metric spaces to topological spaces. If the reason for discussing continuity in analysis is differentiation, then in topology the concept of continuity is needed in order to discuss homeomorphisms.

The following are several useful equivalent conditions for continuous points and continuous functions. Since they are equivalent conditions, depending on the textbook, an equivalent condition may be taken as the definition.

Equivalent Conditions for a Continuous Point

Let $a \in X$. Then the following propositions are equivalent.

• (1): $f : X \to Y$ is continuous at $a$.
• (2): For every $V \in \mathscr{T}_{Y}$ containing $f(a)$, there exists $ U \in \mathscr{T}_{X}$ satisfying $a \in U \subset f^{-1} (V)$.
• (3): For every $\mathcal{N} ( f(a) )$, $f^{-1} ( \mathcal{N} ( f(a) ) )$ is a neighborhood of $a$.
• (4): For every $V \subset Y$ satisfying $f(a) \in V^{\circ}$, $a \in (f^{-1} (V))^{\circ} $

For reference, $\mathcal{N} (a)$ is an open set of $X$ containing $a$, and is called a neighborhood of $a$.

Equivalent Conditions for a Continuous Function

The following propositions are equivalent.

• [1]: $f : X \to Y$ is a continuous function.
• [2]: For every $V \in \mathscr{T}_{Y}$ containing $f(a)$ and every point $a \in f^{-1} (V)$, there exists $ U_{a} \in \mathscr{T}_{X}$ satisfying $a \in U_{a} \subset f^{-1} (V)$.
• [3]: For every open set $V \subset Y$, $f^{-1} (V)$ is an open set in $X$¹.
• [4]: For every closed set $C \subset Y$, $f^{-1} (C)$ is a closed set in $X$.
• [5]: For every $A \subset X$, $f( \overline{A} ) \subset \overline{ f(A) } $
• [6]: There exists a basis $\mathscr{B}$ of $\mathscr{T}_{Y}$ such that $f^{-1} (B) \in \mathscr{T}_{X}$ for every $B \in \mathscr{B}$.
• [7]: There exists a subbasis $\mathscr{S}$ of $\mathscr{T}_{Y}$ such that $f^{-1} (S) \in \mathscr{T}_{X}$ for every $S \in \mathscr{S}$.
• [8] Composition of continuous functions: If $f : X \to Y$ and $g : Y \to Z$ are continuous functions, then the composite function $g \circ f : X \to Z$ is also continuous.

Another Definition of a Continuous Function

In particular, 'theorem [3]: for every open set $V \subset Y$, $f^{-1} (V)$ is an open set in $X$' is used very frequently, and there are many cases where a continuous function is defined outright by [3]. You do not need to memorize all the conditions listed above, but be sure to remember [3] at least, so that you can pull it out and use it at any time.