# Having trouble with this definition of a connected set

• richyw
In summary: The definition I gave you above is the correct definition: a set is connected if it cannot be divided into two non-empty sets, each of which is both open and closed. But maybe your professor doesn't expect you to know this.
richyw

## Homework Statement

My textbook gives me this definition of a connected set.

I have been working through my practice problems and I got to one that asked me to sketch the set given by$$|z+2-i|=2$$and note whether it is open/closed/neither bounded/unbounded and if it is connected

## Homework Equations

$|z-z_a|=r$ describes a circle with centre $z_a$ and radius r

## The Attempt at a Solution

so I have drawn a circle with centre at (-2,i) and radius 2. It contains all of its boundary, so I said it was closed, it can be contained within a circle centred at the origin, so I said it was bounded.

now about the connectedness. The definition my textbook gives me says I need to be able to connect two points in S, using a finite number of lines that are contained in S. The only way I can connect two points in S is by following a circular path. I don't think that a circular path can be made up of a finite number of lines? but apparently this is a connected set. Can someone please explain to me where I am misinterpreting my textbook?

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hi richyw!
richyw said:
now about the connectedness. The definition my textbook gives me says I need to be able to connect two points in S, using a finite number of lines that are contained in S.

ah, that definition starts "an open set is connected if …"

(there's a way of extending the definition to non-open sets, but i don't remember what it is … i suspect it's something to do with being contained in a connected open set)

richyw said:

## Homework Statement

My textbook gives me this definition of a connected set.

I have been working through my practice problems and I got to one that asked me to sketch the set given by$$|z+2-i|=2$$and note whether it is open/closed/neither bounded/unbounded and if it is connected

## Homework Equations

$|z-z_a|=r$ describes a circle with centre $z_a$ and radius r

## The Attempt at a Solution

so I have drawn a circle with centre at (-2,i) and radius 2. It contains all of its boundary, so I said it was closed, it can be contained within a circle centred at the origin, so I said it was bounded.

now about the connectedness. The definition my textbook gives me says I need to be able to connect two points in S, using a finite number of lines that are contained in S. The only way I can connect two points in S is by following a circular path. I don't think that a circular path can be made up of a finite number of lines? but apparently this is a connected set. Can someone please explain to me where I am misinterpreting my textbook?

As it stands now, the definition you have provided in your OP is useless in order to solve the problem. Why? Because it's a definition for when an open set is connected. Like you noted, your set is not open.

So you need to find some definition of connected in your textbook that works for more general sets than open sets.

(Also, I can already see from the definition you provided that your textbook is horrible. No good math book would define connectedness this way. The definition provided is that of polygonal connectedness which happens to coincide with connectedness for open sets. So while it is not technically wrong, it's awful. If you're self-studying from this textbook, then please get another book).

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I am not self-studying. I agree that this book is horrible. It used no formatting whatsoever up until the current edition, which bolds theorems. It has virtually no whitespace between different concepts. It is basically a collection of formulas. I have had to manually box the important ones to memorize, and manually add lines to separate things, like mark end of theorems and proofs and stuff. It's also the most expensive book I've ever seen ($235 for a 400 page book if you buy the north american edition). Although that's a bit off-topic. For now, I don't think that my book mentions "connected set" anywhere else. I think in my advanced calculus course we used a definition that had like unions and stuff in it, but what it basically boiled down to is that you could get from a point in a set, to another point in the set, without leaving it. Is this definition good enough for my exam tomorrow? richyw said: I am not self-studying. I agree that this book is horrible. It used no formatting whatsoever up until the current edition, which bolds theorems. It has virtually no whitespace between different concepts. It is basically a collection of formulas. I have had to manually box the important ones to memorize, and manually add lines to separate things, like mark end of theorems and proofs and stuff. It's also the most expensive book I've ever seen ($235 for a 400 page book if you buy the north american edition).

Although that's a bit off-topic. For now, I don't think that my book mentions "connected set" anywhere else.

I think in my advanced calculus course we used a definition that had like unions and stuff in it, but what it basically boiled down to is that you could get from a point in a set, to another point in the set, without leaving it. Is this definition good enough for my exam tomorrow?

So the book only defines connected for open sets, and then asks in the exercises whether a non-open set is connected? Sigh...

I think you should ask your professor what you should do then. I can give you the definition of connectedness, but I don't know it's what your professor expects.

## What is a connected set?

A connected set is a set in which every pair of points within the set can be joined by a continuous curve without leaving the set.

## What are the conditions for a set to be considered connected?

For a set to be considered connected, it must satisfy two conditions: it must be non-empty and it must not be possible to divide the set into two disjoint non-empty open sets.

## Can a set be connected and disconnected at the same time?

No, a set cannot be both connected and disconnected at the same time. It is either one or the other.

## How is a connected set different from a path-connected set?

A connected set is one in which any two points can be joined by a continuous curve, while a path-connected set is one in which any two points can be joined by a continuous path.

## What is the importance of connected sets in mathematics?

Connected sets play a crucial role in many areas of mathematics, such as topology, analysis, and geometry. They help to define and understand the structure of spaces and provide a foundation for many important theorems and concepts.

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