On continuous and locally one-to-one map

• krete
In summary, the conversation discusses whether a continuous and locally one-to-one map must also be a globally one-to-one map. The answer is no, as shown by the example of the mapping \mathbb{R} \rightarrow S^1 defined by x \mapsto \exp(2\pi i x). It is then asked if this holds true for two open spaces, and the answer is still no, as shown by the counter-example \mathbb{C}\setminus \{0\}\rightarrow \mathbb{C}:z\rightarrow z^2.
krete
Whether a continuous and locally one-to-one map must be a (globally) one-to-one map? If the answer is not. Might you please give a counter-example? Thank in advance.

The answer is no. Consider the mapping $\mathbb{R} \rightarrow S^1$ defined by $x \mapsto \exp(2\pi i x)$.

Got it, many thanks!

Another question: whether a continuous and locally one-to-one map between two open spaces, e.g., two connected open set of R^n, must be a (globally) one-to-one map?

krete said:
Got it, many thanks!

Another question: whether a continuous and locally one-to-one map between two open spaces, e.g., two connected open set of R^n, must be a (globally) one-to-one map?

No, consider

$$\mathbb{C}\setminus \{0\}\rightarrow \mathbb{C}:z\rightarrow z^2$$

Thanks a lot!

1. What is a continuous map?

A continuous map is a function between two topological spaces that preserves the topological structure. This means that small changes in the input will result in small changes in the output. In other words, if points are close together in the domain, their images will also be close together in the range.

2. What does it mean for a map to be locally one-to-one?

A locally one-to-one map is a function that is one-to-one (injective) on small subsets of its domain. This means that for any point in the domain, there is a small open neighborhood around it where the map is one-to-one. In other words, the map does not map multiple points in the domain to the same point in the range.

3. What is the significance of a continuous and locally one-to-one map?

A continuous and locally one-to-one map is important in many areas of mathematics, such as differential geometry and topology. It allows us to study the properties of spaces and their transformations in a more precise and rigorous manner. It also has applications in physics and engineering.

4. How is a continuous and locally one-to-one map different from a continuous and one-to-one map?

A continuous and locally one-to-one map is a stronger condition than a continuous and one-to-one map. The latter only requires the map to be one-to-one on the entire domain, while the former allows for the map to be one-to-one on smaller subsets of the domain. This means that a continuous and locally one-to-one map may not be one-to-one on the entire domain, but it is still locally one-to-one.

5. Can a continuous and locally one-to-one map be surjective?

Yes, a continuous and locally one-to-one map can be surjective. This means that the map can cover the entire range and every point in the range has at least one pre-image in the domain. However, it is not always the case as there are examples of continuous and locally one-to-one maps that are not surjective.

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