Is Every Connected Metric Space Compact?

In summary, the problem from Munkres (Topology) states that a connected metric space with more than one point is uncountable. This can be proven by considering the set of distances between two distinct points in the space and showing that it must be uncountable. This contradicts the assumption that the space is countable and thus proves the original statement. This proof does not require the use of the theorem about nonempty compact Hausdorff spaces.
  • #1
facenian
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Homework Statement


This is a problem from Munkres(Topology): Show that a connected metric space ##M## having having more than one point is uncountable.

Homework Equations


A theorem of that section of the book states: Let ##X## be a nonempty compact Hausdorff space. If no singleton in ##X## is open, then ##X## is uncountable

The Attempt at a Solution

d[/B]
If ##M## is connected and has more than one point then no singleton can be an open set since ##\{x\}\,\text{and}\,X\setminus\{x\}## would be a separation. Then if ##M## were compact application of the above mentioned theorem shows that ##M## is uncountable.
However the problem only states that ##M## is connected.
Is it possible that that every connected metric space be compact? or the problem should be solved without using the above mentioned theorem?
 
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  • #2
No. Certainly not every connected metric space is compact. Take for example ##\mathbb{R}## with the usual topology.

I don't see a way to use the theorem you listed. Here's a hint for a proof:

Let ##x,y## be 2 distinct points in the metric space ##M##. Consider the set ##S :=\{d(x,z): z \in M\}##.

You have to distinguish 2 cases:

(1) S contains the interval ##[0,d(x,y)]##
(2) S does not contain the interval ##[0,d(x,y)]##

The second will lead to a contradiction as ##M## is connected, and the first shows you that ##S## must be uncountable, as it contains an interval.
 
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Related to Is Every Connected Metric Space Compact?

1. What is a metric topology problem?

A metric topology problem involves studying the properties of topological spaces that have a metric defined on them. The goal is to understand how the topology of the space is related to the metric, and how different metrics can induce different topologies on the same set.

2. Why is the metric topology problem important?

The metric topology problem has applications in various fields such as physics, engineering, and computer science. It helps us understand the structure and behavior of spaces, which is crucial in solving real-world problems. Additionally, it provides a framework for understanding continuity, convergence, and other important concepts in analysis and topology.

3. What are some common examples of metric topology problems?

Some common examples of metric topology problems include studying the convergence of sequences and series, understanding the continuity of functions, and analyzing the properties of metric spaces such as completeness, compactness, and connectedness.

4. What are some techniques used to solve metric topology problems?

There are various techniques used to solve metric topology problems, including the use of metric spaces, topological spaces, and various theorems and concepts from analysis and topology. These may include the Hausdorff metric, the Baire category theorem, and concepts such as open and closed sets, neighborhoods, and limit points.

5. Are there any open problems in metric topology?

Yes, there are still many open problems in metric topology. Some of these include the existence of isometries between different metric spaces, the relationship between convergence and continuity, and the classification of different types of metric spaces. These problems continue to be actively studied by mathematicians and have implications in other areas of mathematics and science.

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