Closed set

Closed set
Tag	`0G5AV9Q4RR`
URI	`m2://0G5AV9Q4RR`
Kind	definition
Writing	W1 stated
MSC	54A05
Defines	Closed set
Uses	Topological space, Open set, Complement

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Definition

Let $X$ be a topological space. A subset $F$ of $X$ is closed in $X$ when its complement $X \setminus F$ is open in $X$ .

Remarks

The sets $\emptyset$ and $X$ are both open and closed. A set can be neither open nor closed: $[0, 1)$ in $\mathbb{R}$ is an example. In a metric space, $F$ is closed if and only if it contains the limits of its convergent sequences.

Explanations

L1 human

Closed is not the opposite of open. A closed set is a set whose outside is open. The interval $[0, 1]$ is closed in $\mathbb{R}$ , because its outside $(-\infty, 0) \cup (1, \infty)$ is open.

What this node needs for W2

edge instance_of in: one example (0 of 1)
edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What depends on this

60 nodes use this node, directly or through other nodes.

Closure is the set of limits of sequences 003QP7A8HG
Finite intersection property 00BWV5CKCF
Totally disconnected space 01CYW8S04D
Urysohn's lemma 01EY3VR7ZD
Cantor set is closed 01VT24J1SD
Closure is closed 02TZDKG50Q
Equivalence of norms in finite dimension 03MC70ZRNR
Torus 03NM7DZXXC
Continuous bijection from a compact space to a Hausdorff space is a homeomorphism 0430ETCMKD
T1 space 05GQJXBP3F
Closed subsets of compact sets are compact 066HW6D9A7
Closure in a metric space 068P0P32DW
Heine–Borel theorem 07PFYV65MB
Sequence spaces lp 08P6T2WH6R
Riesz's lemma 097C7KCAT5
Intermediate value theorem for connected spaces 098C0KZZAP
Closed sets contain their limits 09GQ3ZTQJ9
Intersection of closed sets is closed 09YMV8H4X5
Baire category theorem 0A7KGAYHGV
Finite-dimensional normed spaces are complete 0ABP2F2ZP7
Metrizable spaces are normal 0AD0BZ49DW
Normal space 0AJ646N339
Continuity via closed sets 0B8GPXFSP7
Connected subsets of the real line are intervals 0B9ZQFMQNV
Linear maps on finite-dimensional spaces are continuous 0C13E623WD
Dense set 0D198GPMTJ
A set is closed iff it equals its closure 0D2W7FS52T
Baire space 0D5KR33H7X
Connected space 0D939CMME7
Closure 0DRKT1QJR7
Continuous image of a connected set is connected 0EPD87YVG8
Cantor's intersection theorem 0EZBG8SKJM
Closed subsets of complete spaces are complete 0FBXJZX3MK
Meagre set 0FQPYAY9PJ
Completion of a metric space 0G4J7Z7WPJ
Separable space 0GKJ78948V
Regular space 0JF9KW2TXB
Boundary 0JJ9J1XZ8G
Topologist's sine curve 0K3T4A659R
Complement of an open set is closed 0K67Q60Z8P
p-norm on R^n 0KNJWEVSMG
Nowhere dense set 0KSM23SN3H
One-point compactification 0MM45BT6HC
Compact Hausdorff spaces are normal 0MQQQ5NA2S
Open connected subsets of R^n are path-connected 0MXN7AFBPE
Compact subsets of a Hausdorff space are closed 0MYFJVMN7M
Connected components are closed 0NRT0E2QT2
Closed map 0NYZ2M45G9
Heine–Borel theorem in finite-dimensional normed spaces 0PHT6NBYPB
Connected set 0R37DFJKFW
Cantor set is homeomorphic to the space of binary sequences 0RQ8DHHR8V
Closure is the smallest closed superset 0SHC6Z304Z
Continuous real function on a compact set attains its bounds 0V4DET59NW
Compact closed unit ball implies finite dimension 0VZVV8VME5
Complete subspaces are closed 0W41XG4W60
Circle as a quotient of the real line 0WH1GVBJNH
Countable intersection of dense open sets is dense 0X2T63GFQQ
Tietze extension theorem 0X52TNMTCJ
Closed balls are closed 0YS40B38JD
Path-connected spaces are connected 0ZMT1GP4KM

Sources

Web https://en.wikipedia.org/wiki/Closed_set, Definition

What links here

Used by

Finite intersection property 00BWV5CKCF
Urysohn's lemma 01EY3VR7ZD
Cantor set is closed 01VT24J1SD
T1 space 05GQJXBP3F
Closed subsets of compact sets are compact 066HW6D9A7
Heine–Borel theorem 07PFYV65MB
Riesz's lemma 097C7KCAT5
Closed sets contain their limits 09GQ3ZTQJ9
Intersection of closed sets is closed 09YMV8H4X5
Normal space 0AJ646N339
Continuity via closed sets 0B8GPXFSP7
Baire space 0D5KR33H7X
Connected space 0D939CMME7
Closure 0DRKT1QJR7
Cantor's intersection theorem 0EZBG8SKJM
Regular space 0JF9KW2TXB
Complement of an open set is closed 0K67Q60Z8P
One-point compactification 0MM45BT6HC
Compact subsets of a Hausdorff space are closed 0MYFJVMN7M
Closed map 0NYZ2M45G9
Closed balls are closed 0YS40B38JD