Category:Topology Theorems, lemmas and corollaries

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Pages in category "Topology Theorems, lemmas and corollaries"

The following 102 pages are in this category, out of 102 total.

A

A compact and convex subset of Euclidean n-space with non-empty interior is a closed n-cell and its interior is an open n-cell
A compact and convex subset of Euclidean n-space with non-empty interior is a closed n-cell and its interior is an open n-cell/Statement
A continuous map induces a homomorphism between fundamental groups
A continuous map induces a homomorphism on fundamental groups
A function is continuous if and only if the pre-image of every basis element is open
A map is continuous if and only if each point in the domain has an open neighbourhood for which the restriction of the map is continuous on
A map is continuous if and only if the pre-image of every closed set is closed
A proper vector subspace of a topological vector space has no interior
A set is bounded if and only if for all points in the space there is a positive real such that the distance from that point to any point in the set is less than the positive real
A set is dense if and only if every non-empty open subset contains a point of it
A set is open if and only if every point in the set has an open neighbourhood contained within the set
A subset of a topological space is disconnected if and only if it can be covered by two non-empty-in-the-subset and disjoint-in-the-subset sets that are open in the space itself
A subset of a topological space is disconnected if and only if it can be covered by two non-empty-in-the-subset and disjoint-in-the-subset sets that are open in the space itself/Statement
A subset of a topological space is open if and only if it is a neighbourhood to all of its points
A subspace of a Hausdorff space is a Hausdorff space
A subspace of a Hausdorff space is Hausdorff
A topological space is connected if and only if the only sets that are both open and closed in the space are the entire space itself and the emptyset
A topological space is disconnected if and only if it is homeomorphic to a disjoint union of two or more non-empty topological spaces
A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements
An open ball contains another open ball centred at each of its points
An open set is a neighbourhood to all of its points

B

C

C cont.

D

Discrete metric and topology/Summary

E

F

G

H

I

I cont.

Intermediate value theorem

L

Lebesgue number lemma

M

Metric topology

P

R

R^n is a topological vector space

S

Square lemma (of homotopic paths)

T

U

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