on rps-connected spaces
on rps-connected spaces

The Banach-Stone Theorem
The Banach-Stone Theorem

... In metric spaces there is the notion of a (converging) sequence. However, this is not sufficient in general topological spaces. In this section we will discuss a well-known generalization. First of all, we have some definitions. 4.1. Definition. Let (X, T ) be a topological space and x ∈ X. A set N ...
Full text
Full text

Course 3
Course 3

Since Lie groups are topological groups (and manifolds), it is useful
Since Lie groups are topological groups (and manifolds), it is useful

GEOMETRY 5: Set-theoretic topology.
GEOMETRY 5: Set-theoretic topology.

S1-Equivariant K-Theory of CP1
S1-Equivariant K-Theory of CP1

Relations among continuous and various non
Relations among continuous and various non

On πbμ Compactness and πbμ Connectedness in Generalized
On πbμ Compactness and πbμ Connectedness in Generalized

Introductory Analysis 1 The real numbers
Introductory Analysis 1 The real numbers

Weakly sp-θ-closed functions and semipre
Weakly sp-θ-closed functions and semipre

Solutions to exercises in Munkres
Solutions to exercises in Munkres

ON (g, s)-CONTINUOUS AND (πg, s)
ON (g, s)-CONTINUOUS AND (πg, s)

Algebraic Topology Introduction
Algebraic Topology Introduction

p. 1 Math 525 Notes on section 17 Isolated points In general, a point
p. 1 Math 525 Notes on section 17 Isolated points In general, a point

1.6 Smooth functions and partitions of unity
1.6 Smooth functions and partitions of unity

ordered spaces all of whose continuous images are normal
ordered spaces all of whose continuous images are normal

A note on partially ordered topological spaces and a special type of
A note on partially ordered topological spaces and a special type of

TAKS - Lesson 4 - Geneseo Migrant Center
TAKS - Lesson 4 - Geneseo Migrant Center

The Stone-Cech compactification of Tychonoff spaces
The Stone-Cech compactification of Tychonoff spaces

... Theorem 5. Assume that X is a topological space and that fi : X → Xi , i ∈ I, are continuous functions. This family separates points from closed sets if and only if the collection of sets of the form fi−1 (Ui ), i ∈ I and Ui open in Xi , is a base for the topology of X. Proof. Assume that the family ...
Math 396. Gluing topologies, the Hausdorff condition, and examples
Math 396. Gluing topologies, the Hausdorff condition, and examples

SAM III General Topology
SAM III General Topology

SEMI-GENERALIZED CONTINUOUS MAPS IN TOPOLOGICAL
SEMI-GENERALIZED CONTINUOUS MAPS IN TOPOLOGICAL

τ* -Generalized Compact Spaces and τ* -Generalized
τ* -Generalized Compact Spaces and τ* -Generalized

METRIC SPACES AND COMPACTNESS 1. Definition and examples
METRIC SPACES AND COMPACTNESS 1. Definition and examples

Continuous function

In mathematics, a continuous function is, roughly speaking, a function for which small changes in the input result in small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.As an example, consider the function h(t), which describes the height of a growing flower at time t. This function is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous.