WHEN SPACE DOES NOT HAVE DISTANCE: What is the Long Line in Math and Other Examples (Version 2.0)

In many ways metric spaces grant a large amount of structure to a topological space. So it's natural to ask what happens when space does not have distance defined on it. Can we still talk about things like size or even compare these types of spaces to other metrizable spaces? The answer is yes and I'll go through a few examples of such spaces. While answering the question "What is the long line in math" I will also spend time comparing the long line to the real numbers to.

Time Cards for this video:
00:00 Notes on Correction for Version 2.0

01:05 Last time to Metrizability
Although most of the information from last time is helpful and can help give context for this video it's not entirely necessary. I use the last example from the last video to introduce the 'niceness' concept behind metric spaces in terms of all the additional topological structure they endow on a space, it also allows an introduction to metrizability which sling-shots us into the main topic. You can find the video from last time here:    • SEPARATION BUT MATHEMATICALLY: What T...  

03:40 What is the Long Line?
The Long Line in math is a topological space that has a lot of counter intuitive properties of which we will only really focus on one, that the long line is not a metrizable space. In this section however, I go through the definition of the closed and open long ray and their constructions that lead to the definition of the Long Line.

08:26 How different are the long line and the real line?
In this section I'll take a brief but accessible look at one of the primary differences between the Real Line and the Long Line by comparing the size of the two spaces. This answers a question raised earlier in this description, yes, there are cases where you can compare the size of a space that does not have distance and a space that does have such a notion.

09:00 Why does the Long Line not have a sense of distance?
Here I give a visual argument for the inability for one to define the notion of distance (or a metric) on the Long Line. Most rigorous proofs out there solely rely on topological qualities of the Long Line so take this as a grain of intuition and not a proof.

13:29 Other Non-Metrizable examples
Now that we've seen an example where Non-Metrizability cannot be easily remediated, we look at another case of topological spaces that are non-metrizable, but they can be easily manipulated to form metrizable spaces. Pseudometrics are the driving force of both examples in this section.

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This 1st version of this video was originally released on May 4th 2020. It was updated to remedy an error made in the finial section of the video.

Subscriber Count as of initial release: 1,088
Subscriber Count as of Version 2.0 release: 9,119

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