Math 2.0

General topology Continua and dimension theory Theorem

Hahn–Mazurkiewicz theorem

Builds on Hausdorff spaceContinuous functionSurjective functionCompact spaceConnected spaceLocally connected spaceSecond-countable space

Statement

Let XX be a nonempty Hausdorff space. Then there is a continuous surjection [0,1]→X[0, 1] \to X if and only if XX is compact, connected, locally connected and second-countable.

Remarks

A continuous image of [0,1][0, 1] in a Hausdorff space is compact, connected and metrizable, and the map is a quotient map, which carries local connectedness over to XX. For the converse, XX is metrizable by the Urysohn metrization theorem, so it is a locally connected continuum. The Cantor set maps continuously onto every compact metric space. Local connectedness lets one join the images of the ends of each removed interval by small arcs, which extends the map to all of [0,1][0, 1]. For example, the square [0,1]2[0, 1]^2 satisfies the conditions, so there is a space-filling curve onto it. The Warsaw circle is compact, connected and metrizable but not locally connected, so it is not a continuous image of [0,1][0, 1].

Going deeper

For a student

The metric spaces in the theorem are the Peano continua, the locally connected continua. The theorem turns the surprise of the Peano curve into a general rule. It shows that a continuous image of an interval can have any finite dimension. A related fact is that every Peano continuum is arcwise connected.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
Compact space, concept, no evidence levelCompact spaceconceptConnected space, concept, no evidence levelConnected spaceconceptContinuous function, concept, no evidence levelContinuous functionconceptHausdorff space, concept, no evidence levelHausdorff spaceconceptLocally connected space, concept, no evidence levelLocally connected spa…conceptSecond-countable space, concept, no evidence levelSecond-countable spaceconceptSurjective function, concept, no evidence levelSurjective functionconceptHahn–Mazurkiewicz theorem, statement, no evidence levelHahn–Mazurkiewicz the…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Compact space concept
  2. Connected space concept
  3. Continuous function concept
  4. Hausdorff space concept
  5. Locally connected space concept
  6. Second-countable space concept
  7. Surjective function concept
  8. Hahn–Mazurkiewicz theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Space-filling_curve
About this page
TagQ5QFYKD004
URIm2://Q5QFYKD004
Kindstatement, theorem
WritingW0 stub
MSC54F15, 54C05
StatusThis copy of the site has no build data. Run m2 status Q5QFYKD004.
AboutContinuum, Locally connected space
UsesHausdorff space, Continuous function, Surjective function, Compact space, Connected space, Locally connected space, Second-countable space
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Examples