Metric topology

Metric topology
Tag	`0A6YRNZB2R`
URI	`m2://0A6YRNZB2R`
Kind	definition
Writing	W0 stub
MSC	54E35
Defines	Metric topology
Uses	Metric space, Open ball, Topological space

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Definition

Let $(X, d)$ be a metric space. The metric topology of $(X, d)$ is the set $\tau_d$ of all $U \subseteq X$ such that for each $x \in U$ there is a real $\varepsilon > 0$ with $B(x, \varepsilon) \subseteq U$ . Here $B(x, \varepsilon)$ is the open ball with centre $x$ and radius $\varepsilon$ .

Remarks

The set $\tau_d$ is a topology on $X$ : it contains $\varnothing$ and $X$ , and it is closed under unions and finite intersections. Each open ball is in $\tau_d$ , by open balls are open. The open balls form a basis of $\tau_d$ . For $\mathbb{R}$ with the usual metric, $\tau_d$ is the usual topology, generated by the open intervals. Different metrics can give the same topology. Such metrics are equivalent.

Explanations

L1 human

The metric topology is how a metric space becomes a topological space. A set is open when each of its points has a small ball about it inside the set.

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What depends on this

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

Closure is the set of limits of sequences 003QP7A8HG
Finite intersection property 00BWV5CKCF
Compact set 00R0X1WGPX
Totally disconnected space 01CYW8S04D
Urysohn's lemma 01EY3VR7ZD
Isolated point 01PKQYTTMT
Cantor set is closed 01VT24J1SD
Open set 01ZQQDCERP
Open balls form a basis of the metric topology 02EEX96DFS
Closure is closed 02TZDKG50Q
Arzelà–Ascoli theorem 0381S8Z5P0
Equivalence of norms in finite dimension 03MC70ZRNR
Torus 03NM7DZXXC
Lebesgue number lemma 040CFNJEAP
Continuous bijection from a compact space to a Hausdorff space is a homeomorphism 0430ETCMKD
Compact iff complete and totally bounded 054Q5PW5GX
T1 space 05GQJXBP3F
Homeomorphism 05GZ11PWRY
Composition of continuous maps is continuous 05NK26G0DR
Inclusion of a subspace is continuous 065J822CKN
Closed subsets of compact sets are compact 066HW6D9A7
Closure in a metric space 068P0P32DW
Heine–Cantor theorem 06B8BNJCD6
Continuity at a point of a topological space 07N7W5C29X
Heine–Borel theorem 07PFYV65MB
Absolutely convergent series in a Banach space converge 082MGNWNA8
Sequence spaces lp 08P6T2WH6R
Topological embedding 08SVRVHJNY
Riesz's lemma 097C7KCAT5
Intermediate value theorem for connected spaces 098C0KZZAP
Open sets characterised by neighbourhoods 09CJ00H3NW
Closed sets contain their limits 09GQ3ZTQJ9
Completeness of the space of bounded continuous functions 09GRCZVR5E
Metrizable space 09KDHB82SG
Finite intersection of open sets is open 09NDE9V7X1
Intersection of closed sets is closed 09YMV8H4X5
Product of compact sets is compact 09ZKE5XQ74
Open sets of a metric space 0A26X79RTB
Baire category theorem 0A7KGAYHGV
Finite-dimensional normed spaces are complete 0ABP2F2ZP7
Metrizable spaces are normal 0AD0BZ49DW
Normal space 0AJ646N339
Open balls are open 0B2305GNAZ
Continuity via closed sets 0B8GPXFSP7
Connected subsets of the real line are intervals 0B9ZQFMQNV
Banach space 0BDTTM24DP
Linear maps on finite-dimensional spaces are continuous 0C13E623WD
Interior 0C38S0KMJK
Dense set 0D198GPMTJ
A set is closed iff it equals its closure 0D2W7FS52T
Baire space 0D5KR33H7X
Limits are unique in Hausdorff spaces 0D8T984XEE
Connected space 0D939CMME7
Closure 0DRKT1QJR7
Epsilon-delta continuity in metric spaces 0EKGKY5TGH
Continuous image of a connected set is connected 0EPD87YVG8
Locally compact space 0EQMWQ4WMJ
Cantor's intersection theorem 0EZBG8SKJM
Bounded linear maps are continuous 0F3K4C7XYW
Closed subsets of complete spaces are complete 0FBXJZX3MK
Universal property of the quotient topology 0FF2FJ8CH2
Meagre set 0FQPYAY9PJ
Completion of a metric space 0G4J7Z7WPJ
Closed set 0G5AV9Q4RR
Separable space 0GKJ78948V
Interior is the largest open subset 0HAZEFH137
Limit point 0HBQC2GJXT
Union of open sets is open 0HEF46RPEK
First-countable space 0HG11FW8MJ
Equivalent norms 0J3578562S
Regular space 0JF9KW2TXB
Boundary 0JJ9J1XZ8G
Tychonoff's theorem 0JTRSJKCFB
Path-connected space 0K0GE6BCYW
Topologist's sine curve 0K3T4A659R
Complement of an open set is closed 0K67Q60Z8P
Compactness and sequential compactness agree in metric spaces 0K8SGQSV7G
Continuous image of a compact set is compact 0K9AAJ8G51
p-norm on R^n 0KNJWEVSMG
Nowhere dense set 0KSM23SN3H
Universal property of the product topology 0M6W4XNST6
One-point compactification 0MM45BT6HC
Compact Hausdorff spaces are normal 0MQQQ5NA2S
Equicontinuity 0MW2WXTHMC
Hausdorff space 0MXB4CKHFW
Open connected subsets of R^n are path-connected 0MXN7AFBPE
Compact subsets of a Hausdorff space are closed 0MYFJVMN7M
Connected components are closed 0NRT0E2QT2
Open cover 0NVPNC12JY
Closed map 0NYZ2M45G9
Compactness via open covers 0P6PMJB4MA
Metric spaces are Hausdorff 0P9DCDES73
Sup norm on bounded continuous functions 0PARCKE8WP
Heine–Borel theorem in finite-dimensional normed spaces 0PHT6NBYPB
Projections are continuous 0PM88Z9W4M
Interior is open 0PNWCM559P
Compact space 0QPHW703PV
Connected set 0R37DFJKFW
T0 space 0R3B7DX55J
Compact subsets of a metric space are bounded 0R7BTEWCJ2
Cantor set is homeomorphic to the space of binary sequences 0RQ8DHHR8V
Continuous linear map 0S3SMRH0Z5
Path component 0SB9FNZECG
Closure is the smallest closed superset 0SHC6Z304Z
Open map 0SQ2PC596Y
Continuous map of topological spaces 0SY9ECQ19Q
Equivalent metrics 0TWQ5PRKWS
Continuous real function on a compact set attains its bounds 0V4DET59NW
Continuity is equivalent to sequential continuity in metric spaces 0VGPKZ8MRK
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
Continuity via open sets 0YF82GBTND
Closed balls are closed 0YS40B38JD
Proper metric space 0Z4V98VN5E
Path-connected spaces are connected 0ZMT1GP4KM
Operator norm 0ZTEH055YF

Sources

Web https://en.wikipedia.org/wiki/Metric_space, The topology of a metric space

What links here

Used by

Open set 01ZQQDCERP
Open balls form a basis of the metric topology 02EEX96DFS
Metrizable space 09KDHB82SG
Open sets of a metric space 0A26X79RTB
Equivalent norms 0J3578562S
Metric spaces are Hausdorff 0P9DCDES73
Equivalent metrics 0TWQ5PRKWS