Shura-Bora lemma

The Shura-Bora lemma is the name accepted at P. S. Aleksandrov’s scientific school for the following elementary statement of the general topology concerning properties of compact spaces :

Let be

U

${\ displaystyle U}$ $U$ Is an open subset of compact space

X

${\ displaystyle X}$ $X$ , but

\{F_{s}\}_{s\in S}

${\ displaystyle \ {F_ {s} \} _ {s \ in S}}$ ${\ displaystyle \ {F_ {s} \} _ {s \ in S}}$ - some family of closed (and therefore compact) subsets of this space. If a

\bigcap _{s\in S}F_{s}\subset U

${\ displaystyle \ bigcap _ {s \ in S} F_ {s} \ subset U}$ $\ bigcap _ {{s \ in S}} F_ {s} \ subset U$ then there exists a finite set

\{s_{1},s_{2},\dots s_{n}\}\subset S

${\ displaystyle \ {s_ {1}, s_ {2}, \ dots s_ {n} \} \ subset S}$ ${\ displaystyle \ {s_ {1}, s_ {2}, \ dots s_ {n} \} \ subset S}$ such that

\bigcap _{i=1}^{n}F_{s_{i}}\subset U

${\ displaystyle \ bigcap _ {i = 1} ^ {n} F_ {s_ {i}} \ subset U}$ $\ bigcap _ {{i = 1}} ^ {n} F _ {{s_ {i}}} \ subset U$ .

A more concise statement of the Shura-Bora lemma (in terms of non-indexed families of sets):

Let be

U

${\ displaystyle U}$ $U$ Is an open subset of compact space

X

${\ displaystyle X}$ $X$ , but

{\mathcal {F}}

${\ displaystyle {\ mathcal {F}}}$ ${\ mathcal {F}}$ Is a family of closed (and, therefore, compact) subsets of this space, such that

\bigcap {\mathcal {F}}\subset U

${\ displaystyle \ bigcap {\ mathcal {F}} \ subset U}$ $\ bigcap {\ mathcal F} \ subset U$ . Then

\bigcap {\mathcal {F}}_{0}\subset U

${\ displaystyle \ bigcap {\ mathcal {F}} _ {0} \ subset U}$ $\ bigcap {\ mathcal F} _ {0} \ subset U$ for some finite subfamily

{\mathcal {F}}_{0}\subset {\mathcal {F}}

${\ displaystyle {\ mathcal {F}} _ {0} \ subset {\ mathcal {F}}}$ ${\ mathcal F} _ {0} \ subset {\ mathcal F}$ .

To prove the Shura-Bora lemma, it suffices to note that a family consisting of the sets indicated in its formulation $U$ ${\ displaystyle U}$ $U$ and from additions to elements of the family ${\mathcal {F}}$ ${\ displaystyle {\ mathcal {F}}}$ ${\ mathcal {F}}$ is an open space coating $X$ ${\ displaystyle X}$ $X$ and extract the final subcover from this coverage.

The property indicated in the Shura-Bora lemma actually characterizes compact spaces. ^[one]

Generalizations of the Shura-Bora lemma

The Shura-Bora lemma can be generalized to arbitrary (not necessarily compact) spaces, requiring that the family of closed sets considered in it contain at least one compact ^[2] :

Let be

U

${\ displaystyle U}$

U

- an open subset of space

X

{\ displaystyle X}

X

, but

{\mathcal {F}}

{\ displaystyle {\ mathcal {F}}}

{\mathcal {F}}

Is a certain family of closed subsets of this space, at least one of which is compact, and

\bigcap {\mathcal {F}}\subset U

{\ displaystyle \ bigcap {\ mathcal {F}} \ subset U}

\bigcap {\mathcal F}\subset U

. Then

\bigcap {\mathcal {F}}_{0}\subset U

{\ displaystyle \ bigcap {\ mathcal {F}} _ {0} \ subset U}

\bigcap {\mathcal F}_{0}\subset U

for some finite subfamily

{\mathcal {F}}_{0}\subset {\mathcal {F}}

{\ displaystyle {\ mathcal {F}} _ {0} \ subset {\ mathcal {F}}}

{\mathcal F}_{0}\subset {\mathcal F}

.

Under the assumption of Hausdorff property, the Shura-Bora lemma admits the following significant strengthening ^[3] :

Let be

U

${\ displaystyle U}$

U

Is an open subset of Hausdorff space

X

{\ displaystyle X}

X

, but

{\mathcal {F}}

{\ displaystyle {\ mathcal {F}}}

{\mathcal {F}}

Is a family of compact subsets of this space, such that

\bigcap {\mathcal {F}}\subset U

{\ displaystyle \ bigcap {\ mathcal {F}} \ subset U}

\bigcap {\mathcal F}\subset U

. Then there are a finite family

\{\,F_{1},F_{2},\dots ,F_{n}\,\}\subset {\mathcal {F}}

{\ displaystyle \ {\, F_ {1}, F_ {2}, \ dots, F_ {n} \, \} \ subset {\ mathcal {F}}}

\{\,F_{1},F_{2},\dots ,F_{n}\,\}\subset {\mathcal F}

and the final family

\{\,V_{1},V_{2},\dots ,V_{n}\,\}

{\ displaystyle \ {\, V_ {1}, V_ {2}, \ dots, V_ {n} \, \}}

\{\,V_{1},V_{2},\dots ,V_{n}\,\}

open in

X

{\ displaystyle X}

X

sets with the following properties:
but)

F_{i}\subset V_{i}

{\ displaystyle F_ {i} \ subset V_ {i}}

F_{i}\subset V_{i}

for

i=1,2,\dots ,n

{\ displaystyle i = 1,2, \ dots, n}

i=1,2,\dots ,n

;
b)

\bigcap _{i=1}^{n}V_{i}\subset U

{\ displaystyle \ bigcap _ {i = 1} ^ {n} V_ {i} \ subset U}

\bigcap _{{i=1}}^{n}V_{i}\subset U

.

Shura-Bora lemma and compact connected components

The Shura-Bora lemma was fixed as a separate statement with this name in the monographs of P. S. Aleksandrov ^[4] ^[5] , where it was used as an auxiliary to prove the following fundamental theorem, which belongs to M. R. Shure-Bure (1941) ^{[6 ]} :

The connected component of each point of a Hausdorff compact space coincides with its quasicomponent ^[7] .

Some authors call this last theorem also the “Shura-Bora lemma” ^[8] . For the case of metric compacts, it was previously proved by F. Hausdorff (1914) ^[9] .

Notes

↑ Indeed, let some topological space $X$ ${\ displaystyle X}$ possesses the property indicated in the statement of the Shura-Bora lemma. Let us prove that this space is compact. Let be ${\mathcal {V}}$ ${\ displaystyle {\ mathcal {V}}}$ - its arbitrary open cover. Assuming non-empty family ${\mathcal {V}}$ ${\ displaystyle {\ mathcal {V}}}$ choose arbitrary $U\in {\mathcal {V}}$ ${\ displaystyle U \ in {\ mathcal {V}}}$ .
Put ${\mathcal {F}}=\{\,X\setminus V:V\in {\mathcal {V}},\ V\neq U\,\}$ ${\ displaystyle {\ mathcal {F}} = \ {\, X \ setminus V: V \ in {\ mathcal {V}}, \ V \ neq U \, \}}$ ; then $\bigcap {\mathcal {F}}\subset U$ ${\ displaystyle \ bigcap {\ mathcal {F}} \ subset U}$ (insofar as ${\mathcal {V}}$ ${\ displaystyle {\ mathcal {V}}}$ - coverage). Therefore, there is a finite ${\mathcal {F}}_{0}\subset {\mathcal {F}}$ ${\ displaystyle {\ mathcal {F}} _ {0} \ subset {\ mathcal {F}}}$ , for which $\bigcap {\mathcal {F}}_{0}\subset U$ ${\ displaystyle \ bigcap {\ mathcal {F}} _ {0} \ subset U}$ . It is easy to see that the family of open sets consisting of $U$ ${\ displaystyle U}$ and additions to the elements of the family ${\mathcal {F}}_{0}$ ${\ displaystyle {\ mathcal {F}} _ {0}}$ is the final subfamily of the family ${\mathcal {V}}$ ${\ displaystyle {\ mathcal {V}}}$ covering the space $X$ ${\ displaystyle X}$ .
↑ See, for example, R. Engelking. General Topology / Per. with English .. - M .: Mir, 1986. , Corollary 3.1.5 (S. 197).
↑ See, for example, A. Arhangel'skii, M. Tkachenko. Topological groups and related structures . - Atlantis Press, 2008 .-- ISBN 9078677066 . Lemma 2.4.6. This book notes that this statement belongs to topological folklore.
↑ P.S. Aleksandrov and B.A. Pasynkov. Introduction to dimension theory. - M .: Nauka, 1973 .-- S. 171.
↑ P.S. Aleksandrov. Introduction to set theory and general topology. - M .: Nauka, 1977 .-- S. 285.
↑ M.R. Shura-Bura. On the theory of compact spaces. - Mat. Sb., 1941, 9 (51) : 2, 385-388, Theorem I. In this original work, the “Shura-Bora lemma” is not formulated as a separate statement, but is implicitly proved.
↑ The component (connected component) of a point in a topological space is the largest connected subspace of this space containing this point; quasicomponent is the intersection of all open-closed subsets of this space containing a given point. The component of each point of a topological space is contained in its quasicomponent. The converse is generally not true (even in the case of locally compact subspaces of the ordinary Euclidean plane - see Engelking (loc. Cit.), Example 6.1.24), however, in compact sets (i.e., compact Hausdorff spaces), the components of the points coincide with the quasicomponents, as states the indicated theorem. See also its proof in the cited books of P. S. Aleksandrov and R. Engelking.
↑ See, for example, M.V. Keldysh . Feedback on the scientific activities of M.R. Shura-Bura (1968) ; D.K. Musaev . - On the characterization of complete mappings by means of morphisms to zero-dimensional. - Mat. Tr., 7 : 2 (2004), 72–97.
↑ F. Hausdorff. Grundzüge der Mengenlehre. - Leipzig: von Veit, 1914.

[1] Indeed, let some topological space $X$ ${\ displaystyle X}$ possesses the property indicated in the statement of the Shura-Bora lemma. Let us prove that this space is compact. Let be ${\mathcal {V}}$ ${\ displaystyle {\ mathcal {V}}}$ - its arbitrary open cover. Assuming non-empty family ${\mathcal {V}}$ ${\ displaystyle {\ mathcal {V}}}$ choose arbitrary $U\in {\mathcal {V}}$ ${\ displaystyle U \ in {\ mathcal {V}}}$ .
Put ${\mathcal {F}}=\{\,X\setminus V:V\in {\mathcal {V}},\ V\neq U\,\}$ ${\ displaystyle {\ mathcal {F}} = \ {\, X \ setminus V: V \ in {\ mathcal {V}}, \ V \ neq U \, \}}$ ; then $\bigcap {\mathcal {F}}\subset U$ ${\ displaystyle \ bigcap {\ mathcal {F}} \ subset U}$ (insofar as ${\mathcal {V}}$ ${\ displaystyle {\ mathcal {V}}}$ - coverage). Therefore, there is a finite ${\mathcal {F}}_{0}\subset {\mathcal {F}}$ ${\ displaystyle {\ mathcal {F}} _ {0} \ subset {\ mathcal {F}}}$ , for which $\bigcap {\mathcal {F}}_{0}\subset U$ ${\ displaystyle \ bigcap {\ mathcal {F}} _ {0} \ subset U}$ . It is easy to see that the family of open sets consisting of $U$ ${\ displaystyle U}$ and additions to the elements of the family ${\mathcal {F}}_{0}$ ${\ displaystyle {\ mathcal {F}} _ {0}}$ is the final subfamily of the family ${\mathcal {V}}$ ${\ displaystyle {\ mathcal {V}}}$ covering the space $X$ ${\ displaystyle X}$ .

[2] See, for example, R. Engelking. General Topology / Per. with English .. - M .: Mir, 1986. , Corollary 3.1.5 (S. 197).

[3] See, for example, A. Arhangel'skii, M. Tkachenko. Topological groups and related structures . - Atlantis Press, 2008 .-- ISBN 9078677066 . Lemma 2.4.6. This book notes that this statement belongs to topological folklore.

[4] P.S. Aleksandrov and B.A. Pasynkov. Introduction to dimension theory. - M .: Nauka, 1973 .-- S. 171.

[5] P.S. Aleksandrov. Introduction to set theory and general topology. - M .: Nauka, 1977 .-- S. 285.

[6] M.R. Shura-Bura. On the theory of compact spaces. - Mat. Sb., 1941, 9 (51) : 2, 385-388, Theorem I. In this original work, the “Shura-Bora lemma” is not formulated as a separate statement, but is implicitly proved.

[7] The component (connected component) of a point in a topological space is the largest connected subspace of this space containing this point; quasicomponent is the intersection of all open-closed subsets of this space containing a given point. The component of each point of a topological space is contained in its quasicomponent. The converse is generally not true (even in the case of locally compact subspaces of the ordinary Euclidean plane - see Engelking (loc. Cit.), Example 6.1.24), however, in compact sets (i.e., compact Hausdorff spaces), the components of the points coincide with the quasicomponents, as states the indicated theorem. See also its proof in the cited books of P. S. Aleksandrov and R. Engelking.

[8] See, for example, M.V. Keldysh . Feedback on the scientific activities of M.R. Shura-Bura (1968) ; D.K. Musaev . - On the characterization of complete mappings by means of morphisms to zero-dimensional. - Mat. Tr., 7 : 2 (2004), 72–97.

[9] F. Hausdorff. Grundzüge der Mengenlehre. - Leipzig: von Veit, 1914.

Shura-Bora lemma

Generalizations of the Shura-Bora lemma

Shura-Bora lemma and compact connected components

Notes

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