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Inequivalent Topological Spaces

All topological spaces on n points up to homeomorphism for 2 ≤ n ≤ 9 can be found below in two different formats. Assuming X = {0,...,n−1}, each format presents a topology on X as a space-delimited sequence of closures {x}, x = 0,...,n−1, where sets are bit strings of length n (the kth position is 1 iff k is an element of the set).

Alternatively, each line can be interpreted as containing a minimal base for a topology. Whichever interpretation you choose, it must be applied throughout (this is because the dual topologies produced under the two interpretations are not in general homeomorphic).

One format presents the bit strings explicitly, whereas the other presents them as decimal representations of integers obtained by interpreting the bit strings as big-endian binary representations.

T_0 spaces of cardinalities 8 and 9 were obtained from: C. A. Gann and R. A. Proctor, Chapel Hill Poset Atlas http://web.archive.org/web/20190905161049/https://lists-of-posets.math.unc.edu/, v 1.0, July 2005.

Special thanks to Brendan McKay for sending me files containing all inequivalent topologies on 2 through 9 points in late 2019.

The "all_binary" and "all_decimal" files can be downloaded quickest via the zip files at the top. Links to the individual text files follow, including files that only list the T_0 and non-T_0 spaces.

Mark Bowron
April 2021


all_binary.zip
all_decimal.zip


9-spaces_all_binary.txt.gz (363083 spaces)
9-spaces_all_decimal.txt.gz
9-spaces_T_0_binary.txt.gz (183231 spaces)
9-spaces_T_0_decimal.txt.gz
9-spaces_non-T_0_binary.txt.gz (179852 spaces)
9-spaces_non-T_0_decimal.txt.gz
8-spaces_all_binary.txt (35979 spaces)
8-spaces_all_decimal.txt
8-spaces_T_0_binary.txt (16999 spaces)
8-spaces_T_0_decimal.txt
8-spaces_non-T_0_binary.txt (18980 spaces)
8-spaces_non-T_0_decimal.txt
7-spaces_all_binary.txt (4535 spaces)
7-spaces_all_decimal.txt
7-spaces_T_0_binary.txt (2045 spaces)
7-spaces_T_0_decimal.txt
7-spaces_non-T_0_binary.txt (2490 spaces)
7-spaces_non-T_0_decimal.txt
6-spaces_all_binary.txt (718 spaces)
6-spaces_all_decimal.txt
6-spaces_T_0_binary.txt (318 spaces)
6-spaces_T_0_decimal.txt
6-spaces_non-T_0_binary.txt (400 spaces)
6-spaces_non-T_0_decimal.txt
5-spaces_all_binary.txt (139 spaces)
5-spaces_all_decimal.txt
5-spaces_T_0_binary.txt (63 spaces)
5-spaces_T_0_decimal.txt
5-spaces_non-T_0_binary.txt (76 spaces)
5-spaces_non-T_0_decimal.txt
4-spaces_all_binary.txt (33 spaces)
4-spaces_all_decimal.txt
4-spaces_T_0_binary.txt (16 spaces)
4-spaces_T_0_decimal.txt
4-spaces_non-T_0_binary.txt (17 spaces)
4-spaces_non-T_0_decimal.txt
3-spaces_all_binary.txt (9 spaces)
3-spaces_all_decimal.txt
3-spaces_T_0_binary.txt (5 spaces)
3-spaces_T_0_decimal.txt
3-spaces_non-T_0_binary.txt (4 spaces)
3-spaces_non-T_0_decimal.txt
2-spaces_all_binary.txt (3 spaces)
2-spaces_all_decimal.txt
2-spaces_T_0_binary.txt (2 spaces)
2-spaces_T_0_decimal.txt
2-spaces_non-T_0_binary.txt (1 space)
2-spaces_non-T_0_decimal.txt

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