1-manifolds and curves

I have been painfully remiss in keeping this blog up and running lately. I wholeheartedly blame the pretty intense travel schedule I've been on for the last month and a half.

To get back into the game, I start things off with a letter from a reader. Rodolfo Medina write:

Hallo, Michi:

surfing around in internet, looking for an answer to my question, I fell into
your web site.

I'm looking for an answer to the following question:

my intuitive idea is that a one-dimensional connected topological submanifold
of a topological space S should necessarily be the codomain of a curve (if we
define a curve to be a continuous map from an R interval into a topological
space).
Conversely, the codomain of an injective curve, defined in an open R interval,
should necessarily be a one-dimensional topological submanifold of S.
Do you think that's true?, and, if so, how could it be demonstrated? The
difficulty of the first statement is to paste together all charts so to create
a unique homeomorfism.
Thanks for any reply
Rodolfo

So, let's see if we can assemble an answer. I started writing an email answer, which started ballooning way out of control; so after having checked some details with a colleague, I actually have an answer.

The question is in two parts. The first is whether any connected 1-dimensional topological manifold is a curve, viz. an image of an open interval under a continuous map.

This follows since the manifold is second countable, so we can pick a basis for the topology where each piece looks like an open interval, and just glue them together in order to find the curve parametrization.

The second is whether any image of an open interval is a topological 1-manifold.

For this, the answer is no. Consider the map illustrated by the following picture:

Note that this is non-self-intersecting since the loop never really reaches the curve, it only ever comes infinitesimally close. However, since it comes so close, any neighbourhood of the corresponding meeting point will look something like this:

and hence will never be homeomorphic to an interval.

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