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Type | Label | Description |
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Statement | ||
Theorem | cldopn 20101 | The complement of a closed set is open. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.) |
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Theorem | isopn2 20102 | A subset of the underlying set of a topology is open iff its complement is closed. (Contributed by NM, 4-Oct-2006.) |
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Theorem | opncld 20103 | The complement of an open set is closed. (Contributed by NM, 6-Oct-2006.) |
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Theorem | difopn 20104 | The difference of a closed set with an open set is open. (Contributed by Mario Carneiro, 6-Jan-2014.) |
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Theorem | topcld 20105 | The underlying set of a topology is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 3-Oct-2006.) |
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Theorem | ntrval 20106 | The interior of a subset of a topology's base set is the union of all the open sets it includes. Definition of interior of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
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Theorem | clsval 20107* | The closure of a subset of a topology's base set is the intersection of all the closed sets that include it. Definition of closure of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
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Theorem | 0cld 20108 | The empty set is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 4-Oct-2006.) |
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Theorem | iincld 20109* |
The indexed intersection of a collection ![]() ![]() ![]() ![]() |
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Theorem | intcld 20110 | The intersection of a set of closed sets is closed. (Contributed by NM, 5-Oct-2006.) |
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Theorem | uncld 20111 | The union of two closed sets is closed. Equivalent to Theorem 6.1(3) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.) |
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Theorem | cldcls 20112 | A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.) |
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Theorem | incld 20113 | The intersection of two closed sets is closed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.) |
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Theorem | riincld 20114* | An indexed relative intersection of closed sets is closed. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
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Theorem | iuncld 20115* | A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) |
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Theorem | unicld 20116 | A finite union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) |
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Theorem | clscld 20117 | The closure of a subset of a topology's underlying set is closed. (Contributed by NM, 4-Oct-2006.) |
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Theorem | clsf 20118 | The closure function is a function from subsets of the base to closed sets. (Contributed by Mario Carneiro, 11-Apr-2015.) |
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Theorem | ntropn 20119 | The interior of a subset of a topology's underlying set is open. (Contributed by NM, 11-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
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Theorem | clsval2 20120 | Express closure in terms of interior. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
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Theorem | ntrval2 20121 | Interior expressed in terms of closure. (Contributed by NM, 1-Oct-2007.) |
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Theorem | ntrdif 20122 |
An interior of a complement is the complement of the closure. This set
is also known as the exterior of ![]() |
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Theorem | clsdif 20123 | A closure of a complement is the complement of the interior. (Contributed by Jeff Hankins, 31-Aug-2009.) |
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Theorem | clsss 20124 | Subset relationship for closure. (Contributed by NM, 10-Feb-2007.) |
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Theorem | ntrss 20125 | Subset relationship for interior. (Contributed by NM, 3-Oct-2007.) |
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Theorem | sscls 20126 | A subset of a topology's underlying set is included in its closure. (Contributed by NM, 22-Feb-2007.) |
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Theorem | ntrss2 20127 | A subset includes its interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
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Theorem | ssntr 20128 | An open subset of a set is a subset of the set's interior. (Contributed by Jeff Hankins, 31-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.) |
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Theorem | clsss3 20129 | The closure of a subset of a topological space is included in the space. (Contributed by NM, 26-Feb-2007.) |
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Theorem | ntrss3 20130 | The interior of a subset of a topological space is included in the space. (Contributed by NM, 1-Oct-2007.) |
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Theorem | ntrin 20131 | A pairwise intersection of interiors is the interior of the intersection. This does not always hold for arbitrary intersections. (Contributed by Jeff Hankins, 31-Aug-2009.) |
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Theorem | cmclsopn 20132 | The complement of a closure is open. (Contributed by NM, 11-Sep-2006.) |
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Theorem | cmntrcld 20133 | The complement of an interior is closed. (Contributed by NM, 1-Oct-2007.) (Proof shortened by OpenAI, 3-Jul-2020.) |
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Theorem | cmntrcldOLD 20134 | The complement of an interior is closed. (Contributed by NM, 1-Oct-2007.) Obsolete version of cmntrcld 20133 as of 3-Jul-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | iscld3 20135 | A subset is closed iff it equals its own closure. (Contributed by NM, 2-Oct-2006.) |
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Theorem | iscld4 20136 | A subset is closed iff it contains its own closure. (Contributed by NM, 31-Jan-2008.) |
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Theorem | isopn3 20137 | A subset is open iff it equals its own interior. (Contributed by NM, 9-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
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Theorem | clsidm 20138 | The closure operation is idempotent. (Contributed by NM, 2-Oct-2007.) |
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Theorem | ntridm 20139 | The interior operation is idempotent. (Contributed by NM, 2-Oct-2007.) |
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Theorem | clstop 20140 | The closure of a topology's underlying set is entire set. (Contributed by NM, 5-Oct-2007.) |
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Theorem | ntrtop 20141 | The interior of a topology's underlying set is entire set. (Contributed by NM, 12-Sep-2006.) |
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Theorem | 0ntr 20142 | A subset with an empty interior cannot cover a whole (nonempty) topology. (Contributed by NM, 12-Sep-2006.) |
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Theorem | clsss2 20143 | If a subset is included in a closed set, so is the subset's closure. (Contributed by NM, 22-Feb-2007.) |
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Theorem | elcls 20144* | Membership in a closure. Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 22-Feb-2007.) |
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Theorem | elcls2 20145* | Membership in a closure. (Contributed by NM, 5-Mar-2007.) |
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Theorem | clsndisj 20146 | Any open set containing a point that belongs to the closure of a subset intersects the subset. One direction of Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 26-Feb-2007.) |
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Theorem | ntrcls0 20147 | A subset whose closure has an empty interior also has an empty interior. (Contributed by NM, 4-Oct-2007.) |
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Theorem | ntreq0 20148* | Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
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Theorem | cldmre 20149 | The closed sets of a topology comprise a Moore system on the points of the topology. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
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Theorem | mrccls 20150 | Moore closure generalizes closure in a topology. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
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Theorem | cls0 20151 | The closure of the empty set. (Contributed by NM, 2-Oct-2007.) |
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Theorem | ntr0 20152 | The interior of the empty set. (Contributed by NM, 2-Oct-2007.) |
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Theorem | isopn3i 20153 | An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.) |
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Theorem | elcls3 20154* | Membership in a closure in terms of the members of a basis. Theorem 6.5(b) of [Munkres] p. 95. (Contributed by NM, 26-Feb-2007.) (Revised by Mario Carneiro, 3-Sep-2015.) |
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Theorem | opncldf1 20155* | A bijection useful for converting statements about open sets to statements about closed sets and vice versa. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) |
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Theorem | opncldf2 20156* | The values of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) |
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Theorem | opncldf3 20157* | The values of the converse/inverse of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) |
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Theorem | isclo 20158* |
A set ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | isclo2 20159* |
A set ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | discld 20160 | The open sets of a discrete topology are closed and its closed sets are open. (Contributed by FL, 7-Jun-2007.) (Revised by Mario Carneiro, 7-Apr-2015.) |
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Theorem | sn0cld 20161 |
The closed sets of the topology ![]() ![]() ![]() |
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Theorem | indiscld 20162 | The closed sets of an indiscrete topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.) |
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Theorem | mretopd 20163* | A Moore collection which is closed under finite unions called topological; such a collection is the closed sets of a canonically associated topology. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
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Theorem | toponmre 20164 | The topologies over a given base set form a Moore collection: the intersection of any family of them is a topology, including the empty (relative) intersection which gives the discrete topology distop 20066. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
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Theorem | cldmreon 20165 | The closed sets of a topology over a set are a Moore collection over the same set. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
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Theorem | iscldtop 20166* | A family is the closed sets of a topology iff it is a Moore collection and closed under finite union. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
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Theorem | mreclatdemoBAD 20167 | The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 16488. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 6282 update): This proof uses the old df-clat 16409 and references the required instance of mreclatBAD 16488 as a hypothesis. When mreclatBAD 16488 is corrected to become mreclat, delete this theorem and uncomment the mreclatdemo below. |
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Syntax | cnei 20168 | Extend class notation with neighborhood relation for topologies. |
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Definition | df-nei 20169* | Define a function on topologies whose value is a map from a subset to its neighborhoods. (Contributed by NM, 11-Feb-2007.) |
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Theorem | neifval 20170* | The neighborhood function on the subsets of a topology's base set. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
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Theorem | neif 20171 | The neighborhood function is a function of the subsets of a topology's base set. (Contributed by NM, 12-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
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Theorem | neiss2 20172 | A set with a neighborhood is a subset of the topology's base set. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by NM, 12-Feb-2007.) |
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Theorem | neival 20173* | The set of neighborhoods of a subset of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
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Theorem | isnei 20174* |
The predicate "![]() ![]() |
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Theorem | neiint 20175 | An intuitive definition of a neighborhood in terms of interior. (Contributed by Szymon Jaroszewicz, 18-Dec-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
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Theorem | isneip 20176* |
The predicate "![]() ![]() |
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Theorem | neii1 20177 | A neighborhood is included in the topology's base set. (Contributed by NM, 12-Feb-2007.) |
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Theorem | neisspw 20178 | The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.) |
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Theorem | neii2 20179* | Property of a neighborhood. (Contributed by NM, 12-Feb-2007.) |
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Theorem | neiss 20180 |
Any neighborhood of a set ![]() ![]() ![]() ![]() |
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Theorem | ssnei 20181 | A set is included in its neighborhoods. Proposition Viii of [BourbakiTop1] p. I.3 . (Contributed by FL, 16-Nov-2006.) |
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Theorem | elnei 20182 | A point belongs to any of its neighborhoods. Proposition Viii of [BourbakiTop1] p. I.3. (Contributed by FL, 28-Sep-2006.) |
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Theorem | 0nnei 20183 | The empty set is not a neighborhood of a nonempty set. (Contributed by FL, 18-Sep-2007.) |
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Theorem | neips 20184* | A neighborhood of a set is a neighborhood of every point in the set. Proposition of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.) |
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Theorem | opnneissb 20185 | An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.) |
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Theorem | opnssneib 20186 | Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.) |
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Theorem | ssnei2 20187 |
Any subset of ![]() |
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Theorem | neindisj 20188 | Any neighborhood of an element in the closure of a subset intersects the subset. Part of proof of Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.) |
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Theorem | opnneiss 20189 | An open set is a neighborhood of any of its subsets. (Contributed by NM, 13-Feb-2007.) |
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Theorem | opnneip 20190 | An open set is a neighborhood of any of its members. (Contributed by NM, 8-Mar-2007.) |
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Theorem | opnnei 20191* | A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.) |
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Theorem | tpnei 20192 | The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 20189. (Contributed by FL, 2-Oct-2006.) |
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Theorem | neiuni 20193 | The union of the neighborhoods of a set equals the topology's underlying set. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 9-Apr-2015.) |
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Theorem | neindisj2 20194* |
A point ![]() ![]() ![]() ![]() |
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Theorem | topssnei 20195 | A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.) |
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Theorem | innei 20196 | The intersection of two neighborhoods of a set is also a neighborhood of the set. Proposition Vii of [BourbakiTop1] p. I.3 . (Contributed by FL, 28-Sep-2006.) |
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Theorem | opnneiid 20197 | Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.) |
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Theorem | neissex 20198* |
For any neighborhood ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | 0nei 20199 | The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.) |
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Theorem | neipeltop 20200* | Lemma for neiptopreu 20204. (Contributed by Thierry Arnoux, 6-Jan-2018.) |
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