## Counting submanifolds of the plane

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

After thinking about this question and reading this one I am led to ask for an uncountable collection of homeomorphism types of boundaryless connected path-connected submanifolds of the plane.

My guess is that is suffices to consider complements of Cantor sets. However, I do not know how to distinguish ends (up to homeomorphism) sufficiently well to ensure that this works. Are there other, easier, invariants?

-

Given two perfect, totally disconnected subsets $A,B$ of $\mathbb R^2$, is there always a homeomorphism $\varphi$ of $\mathbb R^2$ such that  $\varphi(A)=B$? If the answer is yes, then it is not enough to work with complements of Cantor sets. – Roland Bacher May 17 2010 at 14:29

## 1 Answer

See a theorem of Richards, which implies that homeomorphism types of planar surfaces are in 1-1 correspondence with homeo. types of compact subsets of the Cantor set. I think there should be uncountably many homeo. types of totally disconnected compactums, but I don't know a reference or an argument off the top of my head. I think this should be related to the ordinalities of the accumulation points, but I'm not sure which ordinals can occur.

Addendum: Googling, I found references to a result of Markiewicz-Sierpinski classifying countable compact metric spaces up to homeomorphism by their Cantor-Bendixson rank (see section 3 of this paper for a statement). The CB-rank must be a countable ordinal $\zeta$, and the space is homeomorphic to the ordinal $\omega^\zeta\cdot n+1$ with the order topology for some $n\in \mathbb{N}$. These may all be realized as compact subsets of the line. This gives uncountably many non-homeomorphic compacta, which by Richards' theorem implies that there are uncountably many planar surfaces.

-

Very nice!  I'll take a look at the paper you mention.  – Sam Nead May 18 2010 at 10:59
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 8, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9442843794822693, "perplexity_flag": "head"}