# Continuous functions have closed graphs

• quasar987
In summary, the conversation discusses the closed graph theorem and its converse, which states that a continuous function from one topological space to another has a closed graph. It also mentions that this theorem is equivalent to the second space being Hausdorff. The conversation also mentions other results related to the Hausdorff property.
quasar987
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## Homework Statement

How is the theorem "Continuous functions have closed graphs" proven in the setting of a general topological space? (assuming the theorem is still valid?)

Actually I found a 6 months old thread in which the OP asks help to prove that a continuous map from a topological space X to a Hausdorff space Y has a closed graph

I haven't been able to prove this version either though. I suppose it deals with sequences and the Hausdorff property is there to ensure the uniqueness of a limit, but I'm not familiar enough with sequence theory in general topological spaces to construct an argument.

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I am interested in the converse to the closed graph theorem.

The closed graph theorem says "E and F Banach, and T linear with a closed graph. Then T is continuous."

Very restrictive. Under which conditions is the converse true? I have shown that for a map T btw metric spaces, "T continuous ==> T has a closed graph". But I'm sure it holds under weaker hypotheses!

I just scribbled some stuff down, and I think I managed to prove that if Y isn't Hausdorff, then there exists a topological space X and a continuous map f:X->Y whose graph isn't closed. So, provided what I did was alright (and I think it is - it wasn't a very intricate argument, but based on nets), this means that saying "the graph of any continuous function from X to Y is closed" is equivalent to saying "Y is Hausdorff".

I don't think this is very surprising. There are a few other results of a similar flavor, e.g. a space is Hausdorff iff its diagonal is closed, a Hausdorff space is metrizable iff its diagonal is a zero set, etc.

## What does it mean for a function to have a closed graph?

When we say that a function has a closed graph, it means that the graph of the function is a closed set. This means that the graph contains all of its limit points and does not have any holes or breaks.

## Why is it important for a function to have a closed graph?

A closed graph is important because it guarantees that the function is continuous. This means that there are no abrupt changes or discontinuities in the function, making it easier to analyze and work with mathematically.

## How can you determine if a function has a closed graph?

To determine if a function has a closed graph, you can plot the graph of the function and check for any gaps or breaks. Alternatively, you can use the definition of a closed set to check if the graph contains all of its limit points.

## What are some examples of functions with closed graphs?

Examples of functions with closed graphs include polynomial functions, trigonometric functions, and exponential functions. These functions are continuous and do not have any abrupt changes or discontinuities.

## Can a function have a closed graph but not be continuous?

No, a function cannot have a closed graph and be discontinuous. This is because a closed graph is a necessary condition for continuity. If a function does not have a closed graph, it cannot be continuous.

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