# Homeomorphism between Space and Product

Do there exist examples of non-empty, infinite spaces X not equipped with the discrete topology for with $X \cong X \times X$?

-
I think the answer is no, by dimension reasons if the dimension falls in the interval $(0, infty)$; I think Lebesgue covering dimensions is multiplicative. The answer is no for manifolds, cell complexes or CW-complexes. Maybe you can use, e.g., Kunneth's theorem if there is some non-trivial homology below the top one. But a weaker result is possible; a space $X$ can be homotopically-equivalent to $X \times X$, e.g., for $X=\mathbb R^n ; n< \infty$ –  user99680 Feb 15 '14 at 18:19

Take for example $X=\cup_{n\ge1}\mathbb R^n$ with the topology defined by $U\subset X$ is open if and only if $U\cap\mathbb R^n$ is open for all $n$. It is then easy to check that the map $\phi:X\times X\to X$ defined by $\phi((x_i),(y_i))=(x_1,y_1,x_2,y_2,x_3,\dots)$ is a homeomorphism here an element of $X$ is represented by $(x_i)$ where $(x_i)$ is a sequence of real numbers which is nonzero only for a finite number of $i's$. (also $X=\Pi_{n\ge1}\mathbb R$ with the product topology would also work with the same map defined as above)

For a another example we have a theorem in functional analysis which says that every separable infinite dimentional Hilbert space is isomorphic to $l_2$. Since $l_2\times l_2$ is a separable Hilbert space hence by the theorem it would be isomorphic to $l_2$ (in particular it would be homemorphic)

-

Elementary examples are often zero-dimensional: $\mathbb{Q}$, $\mathbb{P}$ (= the irrationals as a subset of $\mathbb{R}$) and the Cantor set $C \subset [0,1]$ all are homeomorphic to their squares (so even every finite power of itself; even countable power, for the irrationals and $C$.). A trivial example is an infinite space in the indiscrete topology.

Many infinite-dimensional spaces also obey this: $R^\omega$ in the product topology, or any higher power as well, or the Hilbert space $\ell_2$ (not really different, as $R^\omega$ is homeomorphic (as topological spaces) to $\ell_2$).

A nice one-dimensional space due to Erdős: take all points of $\ell_2$ where all coordinates are rational. Erdős showed this space is one-dimensional and it's quite clearly homeomorphic to its square: the even and odd coordinates form copies of the space itself.

-
{"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": 1, "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, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9666244983673096, "perplexity": 147.29005632202788}, "config": {"markdown_headings": true, "markdown_code": true, "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": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435376073161.33/warc/CC-MAIN-20150627033433-00284-ip-10-179-60-89.ec2.internal.warc.gz"}