Let f: X →Y be a function. The graph of f is the subset of X ×Y given by G={(x, f(x)) |x ∈X}. (a

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Let f: X →Y be a function. The graph of f is the subset of X...

Let $f: X \rightarrow Y$ be a function. The graph of $f$ is the subset of $X \times Y$ given by $G=\{(x, f(x)) \mid x \in X\}$. (a) In Exercise 4.10 we established that if $f: X \rightarrow Y$ is continuous and $Y$ is Hausdorff, then $G$ is a closed subset of $X \times Y$. Here we prove a converse. Assume that $X$ and $Y$ are topological spaces and $Y$ is compact. Show that if $G$ is a closed subset of $X \times Y$, then $f$ is continuous. (Hint: Given $U$ open in $Y$, and $x \in f^{-1}(U)$, show that the slice $\{x\} \times Y$ is contained in the open set $(X-G) \cup(X \times U)$, and then apply the Tube Lemma.) (b) Show that the result from part (a) does not hold if we drop the assumption that $Y$ is compact. That is, find an example of a noncompact space $Y$ and a function $f: X \rightarrow Y$ such that the graph of $f$ is a closed subset of $X \times Y$, but $f$ is not continuous.

Let $f: X \rightarrow Y$ be a function. The graph of $f$ is the subset of $X \times Y$ given by $G=\{(x, f(x)) \mid x \in X\}$.
(a) In Exercise 4.10 we established that if $f: X \rightarrow Y$ is continuous and $Y$ is Hausdorff, then $G$ is a closed subset of $X \times Y$. Here we prove a converse. Assume that $X$ and $Y$ are topological spaces and $Y$ is compact. Show that if $G$ is a closed subset of $X \times Y$, then $f$ is continuous. (Hint: Given $U$ open in $Y$, and $x \in f^{-1}(U)$, show that the slice $\{x\} \times Y$ is contained in the open set $(X-G) \cup(X \times U)$, and then apply the Tube Lemma.)
(b) Show that the result from part (a) does not hold if we drop the assumption that $Y$ is compact. That is, find an example of a noncompact space $Y$ and a function $f: X \rightarrow Y$ such that the graph of $f$ is a closed subset of $X \times Y$, but $f$ is not continuous.

Key Concepts

Tube Lemma

The Tube Lemma is a result in topology that applies to product spaces, particularly when one factor is compact. It asserts that for a product space X × Y, if Y is compact and a certain slice around a point is contained in an open set, then there exists a 'tube' or neighborhood around that point in X whose product with Y is also contained in the open set. This lemma is a powerful tool in proving continuity and other properties in product topologies.

Product Topology

The product topology is the topology defined on the Cartesian product of two or more topological spaces, where the open sets are generated by products of open sets from the component spaces. Understanding the product topology is essential when analyzing subsets such as the graph of a function, as it influences how open and closed sets behave in the resulting space.

Hausdorff Spaces

A Hausdorff space is a topological space in which any two distinct points can be separated by disjoint neighborhoods. This separation property is important in various topological arguments, including those ensuring that the graph of a continuous function is closed, because it guarantees the uniqueness of limits.

Continuity in Topological Spaces

Continuity of a function in a topological context is defined by the condition that the preimage of every open set in the codomain is open in the domain. This definition provides a general framework for understanding how functions behave with respect to the topology of their domain and codomain, independent of any metric structure.

Graph of a Function

The graph of a function f: X ? Y is defined as the set of ordered pairs {(x, f(x)) | x ? X} within the product space X × Y. This concept is crucial in topology because it allows one to study the properties of a function by examining the topological characteristics (such as closedness) of its graph in the product space.

Compactness

Compactness is a property of a topological space where every open cover has a finite subcover. This concept is fundamental in topology as it often allows for the extension of properties that hold on finite sets to infinite ones, and it is particularly useful in conjunction with other results such as the Tube Lemma which relies on the compactness of one factor in a product space.

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