A sufficient condition that a monotone image of the three-sphere be a topological three-sphere
Orville G. Harrold · Proceedings of the American Mathematical Society · 1958
1. A continuous transformation of one space onto another is called monotone provided the complete inverse set for each point of the image space is connected. A monotone image of a circle is a simple closed curve or a point. A monotone image of a 2-sphere is a configuration known as a cactoid, i.e. a peano space in which every true cyclic element is a topological 2-sphere. R. L. Moore has shown that if a monotone transformation of a 2-sphere has the additional property that no inverse set separates the 2-sphere, then the image space is again a topological 2-sphere or a point [3]. In the case of the threesphere, S3, as one would expect, the situation is more complicated and extra conditions need to be imposed if the image space is to be expected to look like an S3. A recent example of R. H. Bing [l ] shows that if a monotone transformation on S3 has the property that for each point of the image the complement of the inverse image is an open 3-cell, the image may not be a topological S3, thus answering a long standing conjecture. By studying this example and profiting by conversations with Professor Bing the author was led to the following theorem.