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... open sets are open, preimages of closed sets are closed and preimages of Borel sets are themselves Borel measurable. The situation is more difficult for direct images. That is if f : X → Y is continuous then it does not follow that S ⊆ X being open/closed/measurable implies the same property for f ( ...
... open sets are open, preimages of closed sets are closed and preimages of Borel sets are themselves Borel measurable. The situation is more difficult for direct images. That is if f : X → Y is continuous then it does not follow that S ⊆ X being open/closed/measurable implies the same property for f ( ...
Quotient spaces
... Put more simply, we wish to topologize X/∼ in a way satisfying condition (2). There seems to be no good reason to place any further conditions on what a quotient space should be, so, with this motivation, we make the following definition. Definition. Suppose that X is a topological space on which an ...
... Put more simply, we wish to topologize X/∼ in a way satisfying condition (2). There seems to be no good reason to place any further conditions on what a quotient space should be, so, with this motivation, we make the following definition. Definition. Suppose that X is a topological space on which an ...
the connected and continuity in bitopological spaces 1
... Since Int(U)cl(U) . then Int(Int(U)) Int(cl(U)) and cl(Int(Int(U))) cl(Int(cl(U))) . Therefore U cl(u) cl(Int(cl(U))) . So we have U cl(Int(cl(U))). Hence U is -open. ...
... Since Int(U)cl(U) . then Int(Int(U)) Int(cl(U)) and cl(Int(Int(U))) cl(Int(cl(U))) . Therefore U cl(u) cl(Int(cl(U))) . So we have U cl(Int(cl(U))). Hence U is -open. ...
Finite dimensional topological vector spaces
... Recall that every compact space is also locally compact but there exist locally compact spaces that are not compact such as: • Kd with the euclidean topology • any infinite set endowed with the discrete topology. Indeed, any set X with the discrete topology is locally compact, because for any x ∈ X ...
... Recall that every compact space is also locally compact but there exist locally compact spaces that are not compact such as: • Kd with the euclidean topology • any infinite set endowed with the discrete topology. Indeed, any set X with the discrete topology is locally compact, because for any x ∈ X ...
New examples of totally disconnected locally compact groups
... Every tdlc group G has a compact open subgroup (van Dantzig). An automorphism of a topological group α : G → G is a group isomorphism that is also a homeomorphism (α and α−1 are continuous). If V is a compact open subgroup of G, then α(V) is also compact and open, and α(V) ∩ V is open, so its cosets ...
... Every tdlc group G has a compact open subgroup (van Dantzig). An automorphism of a topological group α : G → G is a group isomorphism that is also a homeomorphism (α and α−1 are continuous). If V is a compact open subgroup of G, then α(V) is also compact and open, and α(V) ∩ V is open, so its cosets ...
(ω)topological connectedness and hyperconnectedness
... is (ω)open. Theorem 12. (X, {Jn }) is maximal (ω)hyperconnected iff it is submaximal and (ω)hyperconnected. Proof. Suppose (X, {Jn }) is maximal (ω)hyperconnected. Let E ⊂ X be (ω)dense. By Corollary 5, (∪n Jn ) − {φ} is an ultrafilter. Therefore E must be (ω)open. For, if E is not (ω)open, then E c ...
... is (ω)open. Theorem 12. (X, {Jn }) is maximal (ω)hyperconnected iff it is submaximal and (ω)hyperconnected. Proof. Suppose (X, {Jn }) is maximal (ω)hyperconnected. Let E ⊂ X be (ω)dense. By Corollary 5, (∪n Jn ) − {φ} is an ultrafilter. Therefore E must be (ω)open. For, if E is not (ω)open, then E c ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.