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In `F33f`_`[algebraic topology`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_topology]`_`f and `F33f`_`[topological data analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Topological_data_analysis]`_`f, the `!Čech complex`! is an `F33f`_`[abstract simplicial complex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Abstract_simplicial_complex]`_`f constructed from a point cloud in any `F33f`_`[metric space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Metric_space]`_`f which is meant to capture `F33f`_`[topological`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Topology]`_`f information about the point cloud or the distribution it is drawn from. Given a finite point cloud `*X`* and an `*ε`* > 0, we construct the Čech complex C ˇ ˇ ε ε ( X ) {\\displaystyle {\\check {C}}_{\\varepsilon }(X)} as follows: Take the elements of `*X`* as the vertex set of C ˇ ˇ ε ε ( X ) {\\displaystyle {\\check {C}}_{\\varepsilon }(X)} . Then, for each σ σ ⊂ ⊂ X {\\displaystyle \\sigma \\subset X} , let σ σ ∈ ∈ C ˇ ˇ ε ε ( X ) {\\displaystyle \\sigma \\in {\\check {C}}_{\\varepsilon }(X)} if the set of `*ε`*-balls centered at points of σ has a `F33f`_`[nonempty`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Empty_set]`_`f `F33f`_`[intersection`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Intersection_(set_theory)]`_`f. In other words, the Čech complex is the `F33f`_`[nerve`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nerve_of_a_covering]`_`f of the set of `*ε`*-balls centered at points of `*X`*. By the `F33f`_`[nerve lemma`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nerve_of_a_covering]`_`f, the Čech complex is homotopy equivalent to the union of the balls, also known as the `F33f`_`[offset filtration`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Offset_filtration]`_`f.`:cite-ref-ghrist-1-0[`F5bf`_`[1`#cite-note-ghrist-1]`_`f]

>>Contents

• `F0af`_`[Relation to Vietoris–Rips complex`#relation-to-vietoris-rips-complex]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>Relation to Vietoris–Rips complex

The Čech complex is a subcomplex of the `F33f`_`[Vietoris–Rips complex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Vietoris–Rips_complex]`_`f. While the Čech complex is more computationally expensive than the Vietoris–Rips complex, since we must check for higher order intersections of the balls in the complex, the nerve theorem provides a guarantee that the Čech complex is homotopy equivalent to union of the balls in the complex. The Vietoris–Rips complex may not be.`:cite-ref-ghrist-1-1[`F5bf`_`[1`#cite-note-ghrist-1]`_`f]

>>See also

• `F33f`_`[Čech cohomology`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Čech_cohomology]`_`f

• `F33f`_`[Computational geometry`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Computational_geometry]`_`f

• `F33f`_`[Simplicial complex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Simplicial_complex]`_`f

• `F33f`_`[Simplicial homology`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Simplicial_homology]`_`f

>>References

`:cite-note-ghrist-1`!1.`! `F0af`_`[↑`#cite-ref-ghrist-1-0]`_`f `:citerefghrist2014`aGhrist, Robert W. (2014). `*Elementary applied topology`* (1st ed.). [United States]. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 9781502880857. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 899283974.`B100`F9d9{{cite book}}`f`b: CS1 maint: location missing publisher (link)

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